®

MATHEMATICS
JEE (MAIN+ADVANCED)
ENTHUSIAST COURSE

EXERCISE
Vector-3D
English Medium
 ALLEN
® Vector

EXERCISE (O-1)
Straight Objective Type
1. Four points A(+1, –1, 1) ; B(1, 3, 1) ; C(4, 3, 1) and D(4, – 1, 1) taken in order are the vertices of
(A) a parallelogram which is neither a rectangle nor a rhombus
(B) rhombus
(C) an isosceles trapezium
(D) a cyclic quadrilateral.
VT0005
2. Let a, b & g be distinct real numbers. The points whose position vector's are a $i + b $j + g k$ ;
b $i + g $j + a k$ and g $i + a $j + b k$

®
(A) are collinear (B) form an equilateral triangle
(C) form a scalene triangle (D) form a right angled triangle
VT0006
3. Let A(0, –1, 1), B(0, 0, 1), C(1, 0, 1) are the vertices of a DABC. If R and r denotes the circumradius

r
and inradius of DABC, then has value equal to
R

3p 3p p p
(A) tan (B) cot (C) tan (D) cot
8 8 12 12
VT0008

( ) ( )
4 . Consider the points A, B and C with position vectors - 2î + 3ˆj + 5k̂ , î + 2 ĵ + 3k̂ and 7 î - k̂
respectively.
r
Statement-1: The vector sum, A B + B C + C A = 0
because
Statement-2: A, B and C form the vertices of a triangle.
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

(A) Statement-1 is true, statement-2 is true and statement-2 is correct explanation for statement-1.
(B) Statement-1 is true, statement-2 is true and statement-2 is NOT the correct explanation for statement-1.
(C) Statement-1 is true, statement-2 is false.
(D) Statement-1 is false, statement-2 is true.
VT0015
5. If the vector 6 $i - 3 $j - 6 k$ is decomposed into vectors parallel and perpendicular to the vector $i + $j + k$
then the vectors are :

( )
(A) - $i + $j + k$ & 7 $i - 2 $j - 5 k$ ( )
(B) - 2 $i + $j + k$ & 8 $i - $j - 4 k$

( )
(C) + 2 $i + $j + k$ & 4 $i - 5 $j - 8 k$ (D) none
VT0018

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6. Let A(1, 2, 3), B(0, 0, 1), C(–1, 1, 1) are the vertices of a DABC.
(i) The equation of internal angle bisector through A to side BC is
r r
(A) r = î + 2ˆj + 3k̂ + µ(3î + 2 ĵ + 3k̂ ) (B) r = î + 2ˆj + 3k̂ + µ(3î + 4 ĵ + 3k̂ )
r r
(C) r = î + 2 ĵ + 3k̂ + µ(3î + 3ˆj + 2k̂ ) (D) r = î + 2 ĵ + 3k̂ + µ(3î + 3ˆj + 4k̂ )

(ii) The equation of median through C to side AB is

r r
(A) r = - î + ˆj + k̂ + p(3î - 2k̂ ) (B) r = - î + ˆj + k̂ + p(3î + 2k̂ )

r r
(C) r = - î + ĵ + k̂ + p( -3î + 2k̂ ) (D) r = - î + ˆj + k̂ + p(3î + 2 ĵ)

(iii) The area (DABC) is equal to

®
9 17 17 7
(A) (B) (C) (D)
2 2 2 2

VT0020
r r r r
7. A, B, C & D are four points in a plane with pv's a , b , c & d respectively such that

(ar - dr )· (br - cr ) = (br - dr )· (cr-ar ) = 0. Then for the triangle ABC, D is its
(A) incentre (B) circumcentre (C) orthocentre (D) centroid
VT0026

r r r r
8. a and b are unit vectors inclined to each other at an angle a, a Î (0, p) and a + b < 1. Then a Î

æ p 2p ö æ 2p ö æ pö æ p 3p ö
(A) ç , ÷ (B) ç , p ÷ (C) ç 0, ÷ (D) ç , ÷
è3 3 ø è 3 ø è 3ø è4 4 ø

VT0027 node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

9. Let a$ , b$ , c$ are three unit vectors such that a$ + b$ + c$ is also a unit vector. If pairwise angles between
a$ , b$ , c$ are q1, q2 and q3 respectively then cos q1 + cos q2 + cos q3 equals
(A) 3 (B) - 3 (C) 1 (D) - 1
VT0029
8
10. A tangent is drawn to the curve y = at a point A (x1 , y1) , where x1 = 2. The tangent cuts the x-axis
x2
® ®
at point B. Then the scalar product of the vectors AB & OB is
(A) 3 (B) - 3 (C) 6 (D) - 6
VT0030

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r r r r r r r r
11. Cosine of an angle between the vectors a + b ( ) ( )
and a - b if | a | = 2, | b | = 1 and a ^ b = 60° is

(A) 37 (B) 9 21 (C) 3 7 (D) none

VT0031
12. The vector equations of two lines L1 and L2 are respectively
r r
r = 17 î - 9 ĵ + 9k̂ + l (3î + ĵ + 5k̂ ) and r = 15î - 8 ĵ - k̂ + m ( 4î + 3 ĵ )

I L1 and L2 are skew lines

II (11, –11, –1) is the point of intersection of L1 and L2
III (–11, 11, 1) is the point of intersection of L1 and L2

®
IV cos–1 3 ( )
35 is the acute angle between L1 and L2

then , which of the following is true?

(A) II and IV (B) I and IV (C) IV only (D) III and IV
VT0034
r r r
13. For some non zero vector V , if the sum of V and the vector obtained from V by rotating it by an
r
angle 2a equals to the vector obtained from V by rotating it by a then the value of a, is

p p 2p 2p
(A) 2np ± (B) np ± (C) 2np ± (D) np ±
3 3 3 3

where n is an integer.
VT0036
r r r r r r r r r
14. Let u, v, w be such that u = 1, v = 2, w = 3 . If the projection of v along u is equal to that of w
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

r r r r r r
along u and vectors v , w are perpendicular to each other then u - v + w equals

(A) 2 (B) 7 (C) 14 (D) 14

VT0037
r r
15. If a and b are non zero, non collinear, and the linear combination
r r r r
(2x - y)a + 4b = 5a + ( x - 2 y)b holds for real x and y then x + y has the value equal to
(A) – 3 (B) 1 (C) 17 (D) 3
VT0038

E 99
JEE-Mathematics ALLEN
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16. Given an equilateral triangle ABC with side length equal to 'a'. Let M and N be two points respectively

AB
on the side AB and AC such that A N = K A C and A M = . If B N and C M are orthogonal
3
then the value of K is equal to

1 1 1 1
(A) (B) (C) (D)
5 4 3 2
VT0039
r r r r r r r r r
17. If p & s are not perpendicular to each other and r x p = qx p & r . s = 0, then r =
r r
r r r æ q . pö r
(A) p . s (B) q + ç r r ÷ p
è p . sø

®
r r
r æ q . sö r r r
(C) q - ç r r ÷ p (D) q + m p for all scalars m
è p . sø
VT0040
r r r r r r
18. If u and v are two vectors such that | u | = 3 ; | v | = 2 and | u ´ v |= 6 then the correct statement is
r r r r
(A) u ^ v Î(0, 90°) (B) u ^ v Î(90°, 180°)
r r r r r r
(C) u ^ v = 90° (D) (u ´ v) ´ u = 6 v
VT0041
® ® ®
19. Given a parallelogram OACB. The lengths of the vectors OA , OB & AB are a, b & c respectively.
® ®
The scalar product of the vectors OC & OB is :

a 2 - 3 b 2 + c2 3 a 2 + b 2 - c2 3 a 2 - b 2 + c2 a 2 + 3 b 2 - c2
(A) (B) (C) (D)
2 2 2 2
VT0042

r r r r r r r r
2p
{( ) ( )}
2
20. Vectors a & b make an angle q = . If a = 1 , b = 2 then a + 3 b x 3 a - b = node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

3
(A) 225 (B) 250 (C) 275 (D) 300
VT0043
® ®
21. If the vector product of a constant vector OA with a variable vector OB in a fixed plane OAB be a
constant vector, then locus of B is :
®
(A) a straight line perpendicular to OA
®
(B) a circle with centre O radius equal to OA
®
(C) a straight line parallel to OA
(D) none of these
VT0044

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r r r
22. The vectors a = $i + 2 $j + 3 k$ ; b = 2 $i - $j + k$ & c = 3 $i + $j + 4 k$ are so placed that the end point of one
vector is the starting point of the next vector. Then the vectors are -
(A) not coplanar
(B) coplanar but cannot form a triangle
(C) coplanar but can form a triangle
(D) coplanar & can form a right angled triangle
VT0046
23. Given the vectors
r
u = 2 î - ˆj - k̂
r
v = î - ĵ + 2k̂

®
r
w = î - k̂
r r r
If the volume of the parallelopiped having – c u , v and c w as concurrent edges, is 8 then 'c' can be
equal to
(A) ± 2 (B) 4 (C) 8 (D) can not be determined
VT0047
Ù
24. Given a = x î + yĵ + 2k̂ , b = $i - $j + k$ , c = i$ + 2$j ; (a b ) = p/2, a × c = 4 then

(A) [a b c] 2 = | a | (B) [a b c] = | a |

(C) [a b c] = 0 (D) [a b c] = | a | 2
VT0048
r r r
25. Let a = a 1 $i + a 2 $j + a 3 k$ ; b = b1 $i + b 2 $j + b 3 k$ ; c = c1 i$ + c2 $j + c3 k$ be three non-zero vectors such
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

2
a1 b1 c1
r r r r r p
that c is a unit vector perpendicular to both a & b . If the angle between a & b is , then a 2 b2 c2 =
6
a3 b3 c3

(A) 0
(B) 1

1
(C) (a 2 + a22 + a32) (b12 + b22 + b32)
4 1

3
(D) (a 2 + a22 + a32) (b12 + b22 + b32) (c12 + c22 + c32)
4 1

VT0049

E 101
JEE-Mathematics ALLEN
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26. A rigid body rotates with constant angular velocity w about the line whose vector equation is,
r
( )
r =l $i + 2 $j + 2 k$ . The speed of the particle at the instant it passes through the point with

p.v. 2î + 3 ĵ + 5k̂ is :

(A) w 2 (B) 2w (C) w 2 (D) none

VT0052
r r r
27. Given 3 vectors V1 = aî + bˆj + ck̂ ; V2 = b î + cˆj + ak̂ ; V3 = cî + aˆj + bk̂
r r r
In which one of the following conditions V1, V2 and V3 are linearly independent?

(A) a + b + c = 0 and a2 + b2 + c2 ¹ ab + bc + ca

®
(B) a + b + c = 0 and a2 + b2 + c2 = ab + bc + ca
(C) a + b + c ¹ 0 and a2 + b2 + c2 = ab + bc + ca
(D) a + b + c ¹ 0 and a2 + b2 + c2 ¹ ab + bc + ca
VT0053
r r r r r r r r
28. Given unit vectors m , n & p such that angle between m & n = angle between p and (m ´ n ) = p 6 ,
rr r
then [ n p m] =

(A) 3 4 (B) 3/4 (C) 1/4 (D) none

VT0054
r
29. The altitude of a parallelopiped whose three coterminous edges are the vectors, A = î + ˆj + k̂ ;
r r r r
B = 2î + 4 ĵ - k̂ & C = î + ĵ + 3k̂ with A and B as the sides of the base of the parallelopiped, is

(A) 2 19 (B) 4 19 (C) 2 38 19 (D) none

node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

VT0056

30. Consider D ABC with A º (a ) ; B º ( b) & C º ( c) . If b . (a + c) = b . b + a . c ; b - a = 3;

® ®
c - b = 4 then the angle between the medians AM & BD is

æ 1 ö æ 1 ö
(A) p - cos-1 ç ÷ (B) p - cos-1 ç ÷
è 5 13 ø è 13 5 ø

æ 1 ö æ 1 ö
(C) cos-1 ç ÷ (D) cos-1 ç ÷
è 5 13 ø è 13 5 ø

VT0057

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EXERCISE (O-2)
Multiple Correct Answer Type
r r r r r r r r r
1. If a , b , c be three non zero vectors satisfying the condition a ´ b = c & b ´ c = a then which of the
following always hold(s) good?
r r r rrr r
(A) a , b , c are orthogonal in pairs [ ]
(B) a b c = b
rrr r r r
[ ]
(C) a b c = c
2
(D) b = c
VT0074
r r r
2. Given the following information about the non zero vectors A, B and C
r r r r r r r r r r
(i) ( A ´ B) ´ A = 0 (ii) B · B = 4 (iii) A · B = -6 (iv) B · C = 6

®
Which one of the following holds good?
r r r r r r r r r r
(A) A ´ B = 0 (B) A · ( B ´ C) = 0 (C) A · A = 8 (D) A · C = -9
VT0075
r r r r
3. If A, B, C and D are four non zero vectors in the same plane no two of which are collinear then
which of the following hold(s) good?
r r r r r r r r
(A) ( A ´ B) · (C ´ D) = 0 (B) ( A ´ C) · ( B ´ D) ¹ 0
r r r r r r r r r r
(C) ( A ´ B) ´ (C ´ D) = 0 (D) ( A ´ C) ´ ( B ´ D) ¹ 0
VT0076
r r r r
4. If a , b , c & d are the pv's of the points A, B, C & D respectively in three dimensional space & satisfy
r r r r r
the relation 3 a - 2 b + c - 2 d = 0 , then :
(A) A, B, C & D are coplanar
(B) the line joining the points B & D divides the line joining the point A & C in the ratio 2 : 1.
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

(C) the line joining the points A & C divides the line joining the points B & D in the ratio 1 : 1
r r r r
(D) the four vectors a , b , c & d are linearly dependent.
VT0077

r é 6 ù r é 2ù r é 3 ù
5. The vectors u = ê - 3ú ; v = ê6 ú ; w = ê 2 ú
êë 2 úû êë 3úû êë- 6úû
(A) form a left handed system
(B) form a right handed system
(C) are linearly independent
(D) are such that each is perpendicular to the plane containing the other two.
VT0078

E 103
JEE-Mathematics ALLEN
®

r r r
6. If a , b , c are non-zero, non-collinear vectors such that a vector

r Ù r r
r
(
p = a b cos 2p - a b ( )) r r r r
( (
c and a vector q = a c cos p - a Ù c )) b then pr + qr is
r r
(A) parallel to a (B) perpendicular to a
r r r r
(C) coplanar with b & c (D) coplanar with a and c
VT0079
7. Which of the following statement(s) hold good ?
r r r r r r r r
(A) if a· b = a· c Þ b = c (a ¹ 0)

r r r r r r r r
(B) if a ´ b = a ´ c Þ b = c (a ¹ 0)

®
r r r r r r r r r r r
(C) if a · b = a · c and a ´ b = a ´ c Þ b = c ( a ¹ 0)
r r r r
r r r r v 2 ´ v3 r v 3 ´ v1
(D) if v1 , v 2 , v3 are non coplanar vectors and k1 = vr ·(vr ´ vr ) ; k 2 = vr · (vr ´ vr )
1 2 3 1 2 3

r r
r v1 ´ v 2 r r r 1
and k 3 = vr ·(vr ´ vr ) then k1·(k 2 ´ k 3 ) = vr · (vr ´ vr )
1 2 3 1 2 3

VT0080
r
8. If the line r = 2 î - ĵ + 3 k̂ + l( î + ĵ + 2 k̂ ) makes angles a, b, g with xy, yz and zx planes respectively
then which of the following are not possible?
(A) sin2a + sin2b + sin2g = 2 & cos2a + cos2b + cos2g = 1
(B) tan2a + tan2b + tan2g = 7 & cot2a + cot2b + cot2g = 5/3
(C) sin2a + sin2b + sin2g = 1 & cos2a + cos2b + cos2g = 2
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

(D) sec2a + sec2b + sec2g = 10 & cosec2a + cosec2b + cosec2g = 14/3

VT0081
9. If a, b, c are different real numbers and a $i + b $j + c k$ ; b $i + c $j + a k$ & c $i + a $j + b k$ are position
vectors of three non-collinear points A, B & C then :

(A) centroid of triangle ABC is

3
(
a +b+c $ $ $
i + j+k )
(B) $i + $j + k$ is equally inclined to the three vectors
(C) perpendicular from the origin to the plane of triangle ABC meet at centroid
(D) triangle ABC is an equilateral triangle.
VT0082

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10. A vector of magnitude 10 along the normal to the curve 3x2 + 8xy + 2y2 – 3 = 0 at its point P(1, 0) can
be

(A) 6î + 8 ĵ (B) -6iˆ + 8jˆ (C) 6î - 8ˆj (D) -6iˆ - 8ˆj
VT0083
11. Let OAB be a regular triangle with side length unity (O being the origin). Also M,N are the points of
r r r
trisection of AB,M being closer to A and N closer to B. Position vectors of A,B,M and N are a, b, m
r
and n respectively. Which of the following hold(s) good ?
r r r 2 1 r r r 5 1
(A) m = xa + yb Þ and y = (B) m = xa + yb Þ x = and y =
3 3 6 6
r r 13 r r 15
(C) m.n equals (D) m.n equals

®
18 18
VT0084
12. Which of the following statement(s) is(are) incorrect ?
r r rr
(A) The relation | (u ´ v) |=| u.v | is only possible if atleast one of the vectors ur and vr is null vector.
r
(B) Every vector contained in the line r(t) = 1 + 2t,1 + 3t,1 + 4t is parallel to the vector 1,1,1 .
r r r r r r
(C) If scalar triple product of three vectors, u, v, w is larger than | u ´ v | then | w |> 1 .

(D) The distance between the x-axis and the line x = y = 1 is 2.

VT0086
r r r
13. Given three vectors U = 2î + 3ˆj - 6k̂ ; V = 6î + 2 ĵ + 3k̂ ; W = 3î - 6 ĵ - 2k̂
r r r
Which of the following hold good for the vectors U, V and W ?
r r r r r r r
(A) U, V and W are linearly dependent (B) ( U ´ V) ´ W = 0
r r r r r r r
(C) U, V and W form a triplet of mutually perpendicular vectors (D) U ´ ( V ´ W ) = 0
VT0087
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

14. Which of the following statement(s) is/are true in respect of the lines
r r r r r r r
r = a + l b; r = c + md where b ´ d ¹ 0
rr
æ | b.d | ö
(A) acute angle between the lines is cos ç r r ÷
-1

è | b || d | ø
r r r r r r
(B) The lines would intersect if [c b d] = [a b d]
r r r r
(C) The lines will be skew if [c - a b d] ¹ 0
r r r r r r
(D) If the lines intersect at r = r0 , then the equation of the plane containing the lines is [r - r0 b d] = 0
VT0088

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r r
15. Let a and b be two non-zero and non-collinear vectors then which of the following is/are always correct
?
r r r rˆˆ r rˆˆ r r r ˆ
(A) a ´ b = [a b i]i + [a b j]j + [a b k]k
rr rˆ rˆ rˆ rˆ rˆ rˆ
(B) a.b = (a.i) (b.i) + (a.j).(b.j) + (a.k) (b.k)
r r r
ˆ ˆ ˆ and vr = aˆ ´ bˆ then | u |=| v |
(C) if u = aˆ - (a.b)b
r r r r r r r r r r r
(D) if c = a ´ (a ´ b) and d = b ´ (a ´ b) then c + d = 0
VT0089
r
16. The value(s) of a Î [0, 2p] for which vector a = ˆi + 3jˆ + (sin 2a ) kˆ makes an obtuse angle with the

®
r a r a
z-axis and the vectors b = (tan a) iˆ - ˆj + 2 sin kˆ and c = (tan a) iˆ + ( tan a ) ˆj - 3 cosec kˆ are
2 2

orthogonal, is/are
(A) tan–1 3 (B) p – tan–1 2 (C) p + tan–1 3 (D) 2p – tan–1 2
VT0192

11
17. The vector ˆi + xjˆ + 3kˆ is rotated through an angle of cos–1 and doubled in magnitude, then it becomes
14

4iˆ + (4x - 2)ˆj + 2kˆ . The value of ' x ' CANNOT be :

2 2 20
(A) - (B) (C) - (D) 2
3 3 17
VT0193
r r
18. The vector c , parallel to the internal bisector of the angle between the vectors a = 7iˆ - 4ˆj - 4kˆ and node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

r r
b = -2iˆ - ˆj + 2kˆ with c = 5 6 , is :

(A)
3
(
5 ˆ ˆ
i - 7 j + 2kˆ ) (B)
3
(
5 ˆ ˆ
i + 7 j - 2kˆ ) (C)
3
(
5 ˆ ˆ
i + 7 j - 2kˆ ) (D)
3
(
5 ˆ ˆ
- i - 7 j + 2kˆ )
VT0194

19. A line passes through a point A with position vector 3iˆ + ˆj - kˆ and is parallel to the vector 2 ˆi - ˆj + 2 kˆ .
If P is a point on this line such that AP = 15 units, then the position vector of the point P is/are

(A) 13 ˆi + 4 ˆj - 9 kˆ (B) 13 ˆi - 4 ˆj + 9 kˆ

(C) 7 ˆi - 6 ˆj + 11kˆ (D) - 7 iˆ + 6 ˆj - 11kˆ VT0195

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20. Which of the followings is/are correct :
r
(A) The angle between t he t wo straight lines r = 3iˆ - 2ˆj + 4kˆ + l ( -2iˆ + ˆj + 2kˆ ) and

r ˆ ˆ ˆ -1 æ 4 ö
r = i + 3j - 2k + m ( 3iˆ - 2ˆj + 6kˆ ) is cos ç ÷
è 21 ø

r r r r r r r
(B) ( r.iˆ ) ( ˆi ´ r ) + ( r.jˆ )( ˆj ´ r ) + ( r.kˆ ) ( kˆ ´ r ) = 0
r
(C) The force determined by the vector r = (1, –8, –7) is resolved along three mutually perpendicular
r
directions, one of which is in the direction of the vector a = 2iˆ + 2ˆj + kˆ . Then the vector component of

r r
the force r in the direction of the vector a is - ( 2iˆ + 2ˆj + kˆ )
7

®
3

1
(D) The cosine of the angle between any two diagonals of a cube is . VT0196
3
21. If the distance between points (a, 5a, 10a) from the point of intersection of the line.
r r
r = ( 2iˆ - ˆj + 2kˆ ) + l ( 2iˆ + 4ˆj + 12kˆ ) and plane r. ( ˆi - ˆj + kˆ ) = 5 is 13 units, then value of a may be

80
(A) 1 (B) –1 (C) 4 (D)
63
VT0197

22. â and b̂ are two given unit vectors at right angle. The unit vector equally inclined with â , b̂ and â ´ bˆ will
be :

1 1
(A) - ( aˆ + bˆ + aˆ ´ bˆ ) (B) ( aˆ + bˆ + aˆ ´ bˆ )
3 3
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

1 1
(C) ( aˆ + bˆ - aˆ ´ bˆ ) (D) - ( aˆ + bˆ - aˆ ´ bˆ ) VT0198
3 3

r r r r r r r r r r 3
23. If a, b, c and d are unit vectors such that (a ´ b).(c ´ d) = l and a . c = , then
2
r r r
(A) a, b, c are coplanar if l = 1
r r
(B) Angle between b and d is 30° if l = –1
r r
(C) angle between b and d is 150° if l = –1
r r
(D) If l = 1 then angle between b and c is 60° VT0199

E 107
JEE-Mathematics ALLEN
®

24. Let p, q, r are non zero vectors and a = p × (q × r) and b = (p × q) ´ r . If a = b then

which of the following holds goods
(A) p and q are orthogonal.

(B) p and r are collinear

(C) q and r are orthogonal

(D) q = l ( p × r ) where 'l' is scalar VT0200

r r r r r r 1
25. If ar = b = cr = d = 1 such that ( ar ´ b ) . ( cr ´ d ) = 1 and a × c = , then which of the following
2
can be correct ?
r r r r r
(A) a × b = 0 (B) a × ( b ´ c ) = 0

®
r r r r r r
(C) a × ( b ´ c ) ¹ 0 (D) a + b + c = 4 + 3 VT0201

node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

108 E
 ALLEN
® Vector

EXERCISE (O-3)
Linked Comprehension Type
Paragraph for Question 1 to 2
Vectors are essential tools for undrstanding and solving problems in physics, engineering and mathematics.
One key aspect of working with vectors is finding the angle between them, which can provide valuable
information about the relationship between the two vectors. The dot product, also known as the scalar
product or inner product, is a crucual operation that allows us to determine the angle between two vectors.
r r
The dot product of two vectors A and B is defined as :
r r r r
A × B = A B cos ( q )
r r
where A and B represent the magnitudes of the vectors A and B, respectively and q is the angle between
the vectors.
From the definition above, we can easily deduce the formula to find the angle between the two vectors

®
:
r r r r
q = arccos éë( A × B) / ( A B ) ùû
®
æ ® Ù ®
ö 2p
1. In a quadrilateral ABCD, A C is the bisector of the ç A B A D÷ which is ,
è ø 3
Ù
| | | | | | æ ö
® ® ® ® ®
15 AC = 3 AB = 5 A D then cos ç BA C D÷ is
è ø
14 21 2 2 7
(A) - (B) - (C) (D)
7 2 7 3 7 14
VT0071
r r r r r r
2. If the two adjacent sides of two rectangles are represented by the vectors p = 5a - 3b ; q = - a - 2b
r r r r r r r 1 r r r
and r = -4 a - b ; s = - a + b respectively, then the angle between the vectors x = ( p + r + s ) and
3
r 1 r r
y = ( r + s)
5
æ 19 ö æ 19 ö
(A) is –cos–1 ç ÷ (B) is cos–1 ç ÷
è 5 43 ø è 5 43 ø
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

æ 19 ö
(C) is p – cos–1 ç ÷ (D) cannot be evaluated
è 5 43 ø
VT0072
Paragraph for Question 3 to 5
r r r r
Consider three vectors p = î + ˆj + k̂ , q = 2î + 4 ĵ - k̂ and r = î + ĵ + 3k̂ and let s be a unit vector, then
r r r
3. p, q and r are
(A) linearly dependent
(B) can form the sides of a possible triangle
r r r
(C) such that the vectors (q - r ) is orthogonal to p
(D) such that each one of these can be expressed as a linear combination of the other two
VT0090

E 109
JEE-Mathematics ALLEN
®

r r r r r r
4. If ( p ´ q ) × r = up + vq + w r , then (u + v + w) equals to
(A) 8 (B) 2 (C) – 2 (D) 4
VT0090
r r r r r r r r r r r r
5. the magnitude of the vector ( p · s )(q ´ r ) + (q · s )( r ´ p) + ( r · s )(p ´ q) is
(A) 4 (B) 8 (C) 18 (D) 2
VT0090
Paragraph for Question 6 to 8
The quadratic equation in y, (a – sin x – sin2x)y2 + (b – cosx – cos2x)y + c = 0 is satisfied by each value
r
of l for which vectors, vr 1 = ˆi + lˆj, vr 2 = lˆi + ˆj + kˆ & v 3 = ˆi + ˆj + lkˆ are linearly dependent. Let l0 Î Q
be one of the values of l.
On the basis of above information, answer the following questions :

®
6. Maximum value of a + b + c is - VT0202
7. For l = l0 value of (vr1 ´ vr 2 ) ´ vr 3 is - VT0203
r r r
8. Number of values of l for which v 1, v 2 & v 3 form a triangle - VT0204
Matrix Match Type

9. List-I List-II
(I) P is point in the plane of the triangle ABC. pv’s of A,B and C are (P) centroid
r r r
a, b and c respectively with respect to P as the origin.
r r r r r r r r
( )( )
If b + c . b - c = 0 and ( c + a ) . ( c - a ) = 0 , then w.r.t. the

triangle ABC,P is its

VT0091
r r r
(II) If a, b, c are the position vectors of the three non collinear points (Q) orthocentre
r uuur uuur uuur
A,B and C respectively such that the vector V = PA + PB + PC node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

is a null vector then w.r.t. the DABC, P is its VT0092

(III) If P is a point inside the DABC such that the vector (R) Incentre
r uuur uuur uuur
( ) ( )
R = (BC)PA + (CA) PB + (AB) PC is a null vector then w.r.t.

the DABC, P is its VT0093

(IV) If P is a point in the plane of the triangle ABC such that the (S) circumcentre
uuur uuur uuur uuur
scalar product PA.CB and PB.AC vanishes, then w.r.t. the
DABC, P is its VT0094
(A) I ® P; II ® Q; III ® R; IV ® S (B) I ® S; II ® P; III ® R; IV ® Q
(C) I ® R; II ® P; III ® Q; IV ® S (D) I ® R; II ® S; III ® Q; IV ® P

110 E
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10. Column-I Column-II

r r r æ p pö
(A) If a , b , c are non coplanar unit vectors, such that (P) ç 4, 2 ÷
è ø
r r
r r r b+c r
a×(b× c ) = , then the angle between ar and b lies in
2

r r r r r æ p 3p ö
(B) Four vectors ar , b , cr , d are such that ( ar × b ) × ( cr × d ) = 0 (Q) ç- 4, 4 ÷
è ø
r
Let P1 and P2 be planes determined by the pair of vectors ar , b
r æ p 5p ö
and cr , d respectively, then angle between P1 and P2 lies in (R) ç 6, 6 ÷
è ø
r r r
If ar and b are two unit vectors such that 2 ar – b and 4 ar + 5 b

®
(C)
r
are perpendicular to each other, then angle between ar and b
lies in
r r r æ p 7p ö
(D) If | ar |= 3, | b |= 5, | cr | = 7 and ar + b + cr = 0 , then angle (S) ç - 6 , 12 ÷
è ø
r
between ar and b lies in VT0205
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-1.p65

E 111
JEE-Mathematics ALLEN
®

EXERCISE (O-4)
Numerical Grid Type
1. A rigid body rotates about an axis through the origin with an angular velocity 10 3 radians/sec.
r
If w points in the direction of $i + $j + k$ and the equation to the locus of the points having tangential speed
20 m/sec. is x2 + y2 + z2 - axy - byz - czx - 2 = 0, then (a + b + c) is equal to
VT0073
r rr r r r
2. Let a, b, c be three non-zero vectors which are pairwise non-collinear. If a + 3b is collinear with c and
r r r r r r
b + 2c is colliner with a , then a + 3b + 6c is : VT0147

r r r ˆi - 2 ˆj r 2iˆ + ˆj + 3kˆ
3. If a and b are vectors in space given by a = and b = , then the value of
5 14

®
r r r
( 2ar + b ) . éë( ar ´ b ) ´ ( ar - 2b )ùû is VT0169
r r r r r r r r r r r r
4. If a, b and c are unit vectors satisfying | a - b |2 + | b - c |2 + | c - a |2 = 9 , then | 2a + 5b + 5c | is
VT0173
r r r r r p r r p r r p r r r
5. Let a = 2, b = 1 and c = 3 such that a ^ b = , b ^ c = and c ^ a = . If a - 2b + c can be
3 4 2
expressed as p + q 2 (where p,q Î I), then (p + q) is VT0206
r r r r r r r r r r r r r r
6. ( )
If p = 3 a - 5 b ; q = 2a + b ; r = a + 4 b ; s = - a + b are four vectors such that sin p Ù q = 1 and
r r r Ù r
( )
sin r Ù s = 1 then 43 cos a b( ) is : VT0070
r r
7. Let â and b̂ be two unit vectors. If the vectors c = aˆ + 2bˆ and d = 5aˆ - 4bˆ are perpendicular to each
other, then the angle between â and b̂ is : VT0148
uuur
8. Two adjacent sides of a parallelogram ABCD are given by AB = 2iˆ + 10 ˆj + 11kˆ and
uuur
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

AD = - ˆi + 2ˆj + 2kˆ . The side AD is rotated by an acute angle a in the plane of the parallelogram so
that AD becomes AD'. If AD' makes a right angle with the side AB, then the cosine of the angle a is
given by - VT0168
r r r r r r r r r r
9. ( ) ( )
Let r = sin x a ´ b + cos y b ´ c + 2 ( c ´ a ) , where a, b & c are non-zero and non-coplanar vectors.

r r r r 4
If r is orthogonal to a + b + c , then minimum value of 2 ( x 2 + y 2 ) is VT0207
p
r r r r r r r r r r r
10. If three points ( 2p - q + 3r ) , ( p - 2q + ar ) and ( bp - 5q ) (where p, q, r are non-coplanar vectors) are

1
collinear, then the value of is VT0208
a+b

112 E
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® Vector

EXERCISE (JM)
uuur uuur
1. If the vectors AB = 3iˆ + 4kˆ and AC = 5iˆ - 2 ˆj + 4kˆ are the sides of a triangle ABC, then the length of the
median through A is : [JEE-MAINS 2013]
(1) 18 (2) 72 (3) 33 (4) 45
VT0150
r r r r r
2. ˆ b = iˆ + 2ˆj - kˆ and c = ˆi + ˆj - 2kˆ be three vectors. A vectors of the type b + l c for some
Let a = 2iˆ - ˆj + k,

r 2
scalar l, whose projection on a is of magnitude , is : [JEE-MAINS Online 2013]
3

(1) 2iˆ + 3jˆ - 3kˆ (2) 2iˆ + ˆj + 5kˆ (3) 2iˆ - ˆj + 5kˆ (4) 2iˆ + 3jˆ + 3kˆ

®
VT0151
r r r r r r
3. Let ar = 2iˆ + ˆj - 2kˆ , b = ˆi + ˆj . If cr is a vector such that a · c = c , c - a = 2 2 and the angle between
r
r r r
a × b and c is 30º, then ( ar ´ b ) ´ cr equals : [JEE-MAINS Online 2013]

3 1 3 3
(1) (2) 3 (3) (4)
2 2 2
VT0152
r rr rr r rrr 2
4. If éëa ´ b b ´ c c ´ a ùû = l éëa b c ùû then l is equal to : [JEE(Main)-2014]

(1) 2 (2) 3 (3) 0 (4) 1

VT0153
r r
5. Let a , b and cr be three non-zero vectors such that no two of them are collinear and
r r r r r
( ar ´ b ) ´ cr = 13 b c a . If q is the angle between vectors b and cr , then a value of sin q is :

[JEE(Main)-2015]
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

2 -2 3 2 2 - 2
(1) (2) (3) (4)
3 3 3 3
VT0154

Let a, b and c be three unit vectors such that a ´ æç b ´ c ö÷ =

® ® ® ® ® ® 3 æ ® ®ö ® ®
6. ç b + c ÷ . If b is not parallel to c , then
è ø 2 è ø
® ®
the angle between a and b is :- [JEE(Main)-2016]

5p 3p p 2p
(1) (2) (3) (4)
6 4 2 3
VT0155

E 113
JEE-Mathematics ALLEN
®

r r r r r
7. Let a = 2iˆ + ˆj - 2kˆ and b = ˆi + ˆj . Let cr be a vector such that | cr - ar | = 3, (a ´ b) ´ c = 3 and the angle
r r r r
between cr and a ´ b be 30º. Then a·c is equal to : [JEE(Main)-2017]

1 25
(1) (2) (3) 2 (4) 5
8 8

VT0156
r r
Let u be a vector coplanar with the vectors ar = 2iˆ + 3ˆj - kˆ and b = ˆj + kˆ . If u is perpendicular to ar
r
8.

rr r2
and u.b = 24, then u is equal to - [JEE(Main)-2018]
(1) 315 (2) 256 (3) 84 (4) 336
VT0157

®
r r r r r r r r r
9. Let a = ˆi - ˆj, b = ˆi + ˆj + kˆ and rc be a vector such that a ´ c + b = 0 and a.c = 4, then c 2 is equal to :-

[JEE(Main) 19]

19 17
(1) (2) 8 (3) (4) 9
2 2

VT0158

10. Let 3iˆ + ˆj, ˆi + 3jˆ and bˆi + (1 - b)ˆj respectively be the position vectors of the points A, B and C with
respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB

3
is , then the sum of all possible values of b is :- [JEE(Main)-19]
2

(1) 2 (2) 1 (3) 3 (4) 4

VT0159
r r r r r r
11. Let ar = 3iˆ + ˆj and b = 2iˆ - ˆj + 3kˆ . If b = b1 - b2 , where b1 is parallel to ar and b2 is perpendicular to node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

r r r
a , then b1 ´ b2 is equal to [JEE(Main)-19]

(1) -3iˆ + 9ˆj + 5kˆ (2) 3iˆ - 9ˆj - 5kˆ (3)

1
2
(
-3iˆ + 9ˆj + 5kˆ ) (4)
2
(
1 ˆ ˆ
3i - 9 j + 5kˆ )
VT0160

12. The distance of the point having position vector -ˆi + 2ˆj + 6kˆ from the straight line passing through

the point (2, 3, –4) and parallel to the vector, 6iˆ + 3jˆ - 4kˆ is : [JEE(Main)-19]

(1) 7 (2) 4 3 (3) 2 13 (4) 6

VT0161

114 E
 ALLEN
® Vector
r r r r r r r rr
13. Let ar , b and cr be three units vectors such that ar + b + cr = 0 . If l = a ·b + b·c + c·a and
r r r r r r r r
d = a ´ b + b ´ c + c ´ a , then the ordered pair, (l,d) is equal to : [JEE(Main)-20]

æ 3 r r ö æ 3 r r ö æ 3 r r
ö æ 3 r r ö
(1) ç - ,3a ´ b ÷ (2) ç - ,3c ´ b ÷ (3) ç ,3b ´ c ÷ (4) ç ,3a ´ c ÷
è 2 ø è 2 ø è2 ø è2 ø
VT0162
r r r
14. Let ar = ˆi - 2ˆj + kˆ and b = $i - ˆj + kˆ be two vectors. If cr is a vector such that b ´ cr = b ´ ar and cr × ar = 0 ,
r
then cr × b is equal to [JEE(Main)-20]

1 1 3
(1) (2) –1 (3) - (4) -
2 2 2
VT0163

®
15. Let the volume of a parallelopiped whose coterminous edges are given by ur = ˆi + ˆj + lk,
ˆ vr = ˆi + ˆj + 3kˆ
r r r
and w = 2iˆ + ˆj + kˆ be 1 cu. unit. If q be the angle between the edges u and w , then cosq can be
[JEE(Main)-20]
7 5 7 5
(1) (2) (3) (4)
6 3 7 6 6 3 3
VT0164

r r
r r r r r r r p
16. Let a, b and c be three vectors such that a = 3 , b = 5 , b . c = 10 and the angle between b and c is . If
3
r r r r r r
a is perpendicular to the vector b ´ c , then a ´ ( b ´ c ) is equal to _____. [JEE(Main)-20]
VT0165
r
17. If the vectors, p = (a + 1)iˆ + ajˆ + akˆ , [JEE(Main)-20]
r
q = aiˆ + (a + 1)ˆj + akˆ and
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

r ˆ ˆ rr 2 r r2
r = ai + aj + (a + 1)kˆ (a Î R) are coplanar and 3 ( p.q ) - l r ´ q = 0 , then the value of l is ———.

VT0166
18. The projection of the line segment joining the points (1, –1, 3) and (2, –4, 11) on the line joining the
points (–1, 2, 3) and (3, –2, 10) is ———. [JEE(Main)-20]
VT0167
uuur uuur uuur
19. Let O be the origin. Let OP = xiˆ + yjˆ - kˆ and OQ = - ˆi + 2ˆj + 3xkˆ , x, y Î R, x > 0, be such that PQ = 20
uuur uuur uuur uuur uuur
and the vector OP is perpendicular to OQ . If OR = 3iˆ + zjˆ - 7kˆ , z Î R, is coplanar with OP and OQ ,
2 2 2
then the value of x + y + z is equal to [JEE(Main)-21]
(1) 7 (2) 9 (3) 2 (4) 1
VT0209

E 115
JEE-Mathematics ALLEN
®

20. A hall has a square floor of dimension 10m × 10m (see the figure) and vertical walls. If the angle GPH
1
between the diagonals AG and BH is cos -1 then the height of the hall (in meters) is :
5
[JEE(Main)-21]

E F

H G

P
A B

D C

®
(1) 5 (2) 2 10 (3) 5 3 (4) 5 2
VT0210
r r r r r r r r r
21. (
Let a = ˆi + ˆj + 2kˆ and b = - ˆi + 2ˆj + 3kˆ . Then the vector product ( a + b ) ´ ( a ´ ( ( a - b ) ´ b ) ) ´ b is )
equal to : [JEE(Main)-21]
(1) 5 ( 34iˆ - 5ˆj + 3kˆ ) (2) 7 ( 34iˆ - 5ˆj + 3kˆ ) (3) 7 ( 30iˆ - 5ˆj + 7kˆ ) (4) 5 ( 30iˆ - 5ˆj + 7kˆ )
VT0211
r r r r r r r r r r r r
22. Let a, b, c be three non-coplanar vectors such that a ´ b = 4c , b ´ c = 9a and c ´ a = ab , a > 0.

r r r 1
If a + b + c = , then a is equal to ______. [JEE(Main)-22]
36
VT0212
r r r
23. Let the vectors a = (1 + t ) ˆi + (1 - t ) ˆj + kˆ , b = (1 - t ) ˆi + (1 + t ) ˆj + 2kˆ and c = tiˆ - tjˆ + kˆ , t Î R be such
r r r r node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

that for a, b, g Î R , aa + b b + gc = 0 Þ a = b = g = 0 . Then, the set of all values of t is :

[JEE(Main)-22]
(A) a non-empty finite set (B) equal to N
(C) equal to R – {0} (D) equal to R VT0213
p
24. Let â and b̂ be two unit vectors such that the angle between them is . If q is the angle between
4

the vectors ( â + bˆ ) and ( aˆ + 2bˆ + 2 ( aˆ ´ bˆ ) ) , then the value of 164 cos q is equal to :
2

[JEE(Main)-22]
(A) 90 + 27 2 (B) 45 + 18 2 (C) 90 + 3 2 (D) 54 + 90 2
VT0214

116 E
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EXERCISE (JA)
uuur uuur
1. Let PR = 3iˆ + ˆj - 2kˆ and SQ = ˆi - 3jˆ - 4kˆ determine diagonals of a parallelogram PQRS and
uuur uuur uuur
PT = ˆi + 2ˆj + 3kˆ be another vector. Then the volume of the parallelepiped determined by the vectors PT, PQ
uur
and PS is [JEE-Advanced 2013, 2M]
(A) 5 (B) 20 (C) 10 (D) 30
VT0175

2. { }
Consider the set of eight vectors V = aiˆ + bjˆ + ckˆ : a, b, c Î{-1,1} . Three non-coplanar vectors can

be chosen from V in 2p ways. Then p is [JEE-Advanced 2013, 4, (–1)]

VT0176

®
3. Match List-I with List-II and select the correct answer using the code given below the lists.
List-I List-II
r
r and
P. Volume of parallelepiped determined by vectors a, b 1. 100
r
c is 2. Then the volume of the parallelepiped determined by
r r r r r r
( ) ( )
vectors 2 a ´ b ,3 b ´ c and ( c ´ a ) is

r r r
Q. Volume of parallelepiped determined by vectors a, b and c 2. 30
is 5. Then the volume of the parallelepiped determined by
r r r r r r
( )( )
vectors 3 a + b , b + c and 2 ( c + a ) is

R. Area of a triangle with adjacent sides determined by vectors 3. 24

r r
a and b is 20. Then the area of the triangle with adjacent
r r r r
( )
sides determined by vectors 2a + 3b and a - b ( ) is
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

S. Area of a parallelogram with adjacent sides determined by 4. 60

r r
vectors a and b is 30. Then the area of the parallelogram
r r r
( )
with adjacent sides determined by vectors a + b and a is

Codes :
P Q R S
(A) 4 2 3 1
(B) 2 3 1 4
(C) 3 4 1 2
(D) 1 4 3 2 [JEE-Advanced 2013, 3, (–1)]
VT0177
E 117
JEE-Mathematics ALLEN
®

r r r
4. Let x, y and z be three vectors each of magnitude 2 and the angle between each pair of them is

p r r r r r
. If a is a nonzero vector perpendicular to x and y ´ z and b is nonzero vector perpendicular to
3
r r r
y and z ´ x , then [JEE(Advanced)-2014, 3]
r r r r r r r r r r r r r r r r r r r r r
(A) b = (b.z) (z - x) (B) a = (a.y)(y - z) (C) a .b = -(a .y) (b.z) (D) a = (a.y)(z - y)
VT0178
r r r
5. Let a,b , and c be three non-coplanar unit vectors such that the angle between every pair of them

p r r r r r r r p 2 + 2q 2 + r 2
is . If a ´ b + b ´ c = pa + qb + rc , where p,q and r are scalars, then the value of is
3 q2

®
[JEE(Advanced)-2014, 3]
VT0179
r uuur r uuur r uuur r rr r
6. Let DPQR be a triangle. Let a = QR, b = RP and c = PQ . If | a |= 12, | b |= 4 3 and b.c = 24 , then
which of the following is (are) true ? [JEE 2015, 4M, –2M]
r r
| c |2 r | c |2 r r r r r rr
(A) - | a |= 12 (B) + | a |= 30 (C) | a ´ b + c ´ a |= 48 3 (D) a.b = -72
2 2
VT0180
r r r r
Suppose that p, q and r are three non-coplanar vectors in ¡ . Let the components of a vector s
3
7.
r r r r r r
along p, q and r be 4,3 and 5, respectively. If the components of this vector rs along ( -p + q + r ) ,
r r r r r r
(p - q + r ) and ( - p - q + r ) are x,y and z, respectively, then the value of 2x + y + z is
[JEE 2015, 4M, –0M]
VT0181
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

1 ˆ ˆ ˆ . Given that there exists a

8. Let û = u1ˆi + u 2 ˆj + u 3 kˆ be a unit vector in ¡3 and ŵ = (i + j + 2k)
6
r r r
vector v in ¡3 such that û ´ v = 1 and w.(u
ˆ ˆ ´ v) = 1 . Which of the following statement(s) is(are)

correct?
r
(A) There is exactly one choice for such v
r
(B) There are infinitely many choice for such v
(C) If û lies in the xy-plane then | u1 | = | u 2 |

(D) If û lies in the xz-plane then 2 | u1 | = | u 3 | [JEE(Advanced)-2016, 4(–2)]

VT0182

118 E
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9. Let O be the origin and let PQR be an arbitrary triangle. The po int S is such that
uuur uuur uuur uuur uuur uuur uuur uuur uuur uuur uuur uuur
OP.OQ + OR.OS = OR.OP + OQ.OS = OQ.OR + OP.OS . Then the triangle PQR has S as its
[JEE(Advanced)-2017]
(A) incentre (B) orthocenter (C) circumcentre (D) centroid
VT0183
PARAGRAPH :
uuur uuur uuur uuur uuur uuur
Let O be the origin, and OX, OY, OZ be three unit vectors in the directions of the sides QR, RP, PQ ,
respectively, of a triangle PQR. [JEE(Advanced)-2017]
uuur uuur
10. OX ´ OY =

®
(A) sin(Q + R) (B) sin(P + R) (C) sin 2R (D) sin(P + Q)
VT0184
11. If the triangle PQR varies, then the minimum value of cos(P + Q) + cos(Q + R) + cos(R + P) is

3 3 5 5
(A) (B) - (C) (D) -
2 2 3 3
VT0184
r r r r r r r r r
12. Let a and b be two unit vectors such that a.b = 0 . For some x, y Î ¡ , let c = xa + yb + (a ´ b) .
r r r r
If | c | = 2 and the vector c is inclined at the same angle a to both a and b , then the value of

8cos2 a is ______ [JEE(Advanced)-2018, 3(0)]

VT0185
r
13. Three lines L 1 : r = lˆi, l Î ¡ ,
r r
L 2 : r = k + mˆj, m Î ¡ and
r
L 3 : r = ˆi + ˆj + vk,
ˆ n Î ¡
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

are given. For which point(s) Q on L2 can we find a point P on L1 and a point R on L3 so that
P, Q and R are collinear ? [JEE(Advanced)-2019, 4(–1)]

1 1
(1) k̂ + ˆj (2) k̂ (3) k̂ + ˆj (4) k̂ - ˆj
2 2
VT0186
r r r r r
14. Let a = 2iˆ + ˆj - kˆ and b = ˆi + 2 ˆj + kˆ be two vectors. Consider a vector c = aa + b b , a,b Î ¡. If

r r r r r r r
( ) ( ( ))
the projection of c on the vector a + b is 3 2 , then the minimum value of c - a ´ b .c equals

[JEE(Advanced)-2019, 3(0)]
VT0187

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JEE-Mathematics ALLEN
®

r r r r r
r uuuv r uuuv r uuuv r a.(c – b) |a|
15. In a triangle PQR, let a = QR, b = RP and c = PQ . If a = 3, b = 4 and r r r = r r ,
c.(a - b) |a|+|b|

r r2
then the value of a ´ b is ______ [JEE(Advanced)-2020]

VT0188
uuur uur
16. Let a and b be positive real numbers. Suppose PQ = aiˆ + bjˆ and PS = aiˆ - bjˆ are adjacent sides
r r r uuur uur
of a parallelogram PQRS. Let u and v be the projection vectors of w = ˆi + ˆj along PQ and PS ,
r r r
respectively. If u + v = w and if the area of the parallelogram PQRS is 8, then which of the following
[JEE(Advanced)-2020]

®
statements is/are TRUE ?
(A) a + b = 4
(B) a – b = 2
(C) The length of the diagonal PR of the parallelogram PQRS is 4
r uuur uur
(D) w is an angle bisector of the vectors PQ and PS VT0189
r r r be vectors in three-dimensional space, where r and r are unit vectors which are
17. Let u, v and w u v
r r r r r r
not perpendicular to each other and u.w = 1, v.w = 1, w.w = 4 .
r r r,
If the volume of the parallelopiped, whose adjacent sides are represented by the vectors u, v and w
r r
is 2 , then the value of 3u + 5v is____. [JEE(Advanced)-2021]
VT0190

uuur uuur uuur 1 uuur uuur

18. ˆ OB = ˆi - 2ˆj + 2kˆ and OC = ( OB – l OA ) for some l > 0. If
Let O be the origin and OA = 2iˆ + 2ˆj + k,
2

uuur uuur 9 node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

OB ´ OC = , then which of the following statements is (are) TRUE ? VT0191

2

uuur uuur 3
(A) Projection of OC on OA is - [JEE(Advanced)-2021]
2

9
(B) Area of the triangle OAB is
2

9
(C) Area of the triangle ABC is
2

uuur uuur p
(D) The acute angle between the diagonals of the parallelogram with adjacent sides OA and OC is
3
120 E
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19. Let ˆi, ˆj and k̂ be the unit vectors along the three positive coordinate axes. Let
r
a = 3iˆ + ˆj - k,
ˆ
r
b = ˆi + b 2 ˆj + b 3 k,
ˆ b2, b3 Î R,
r
c = c1 ˆi + c 2 ˆj + c3 k,
ˆ c1, c2, c3 Î R
r
be three vectors such that b2b3 > 0, ar × b = 0 and

æ 0 -c 3 c 2 öæ 1 ö æ 3 - c1 ö
ç ÷ç ÷ ç ÷
ç c3 0 -c1 ÷ç b2 ÷ = ç 1 - c 2 ÷ .
ç -c 0 ÷ç ÷ ç ÷
è 2 c1 øè b3 ø è -1 - c3 ø

®
Then, which of the following is/are TRUE ? [JEE(Advanced)-2022]
r r r
(A) ar × cr = 0 (B) b × cr = 0 (C) b > 10 (D) c £ 11
VT0215
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

E 121
JEE-Mathematics ALLEN
®

ANSWER KEY
Do yourself - 1

p p
1. 7 2 A 3. a = 2np ± and q = 2np +
3 6

Do yourself - 2
4. C 5. C 6. A

Do yourself - 3
r r r r r r r r r r r r r r r r r r r r
1. (a) a, d; b, x, z; c, y (b) b, x;a, d; c, y (c) a, y, z (d) b, z; x, z

®
1
2. 3. A,B,C 4. A 5. A,B,C,D
4

Do yourself - 4
r r
12a - 13b r 3ˆ 6ˆ 2 ˆ
1. , -5b 2. i - j+ k 3. B 4. A
5 7 7 7

5. B 6. D 7. D 8. B 9. B

10. C 11. r r r r
a+b+c =0

Do yourself - 5
2. B 3. A 4. B 5. B 6. B
r ˆ ˆ ˆ + l(3iˆ - 2 ˆj + 6k)
ˆ
7. r = (i - 2 j + 3k)
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

Do yourself - 6

p
1. 2. –15 3. 1
6

2 2 ˆ ˆ 190iˆ - 110ˆj + 45kˆ 2 ˆ 32 ˆ 8 ˆ

4. , ˆ and
(3i + 6 j + 2k) 5. - i+ j- k
7 49 49 21 21 21

6. C 7. B 8. D 9. C 10. D
11. D 12. D 13. D 14. A 15. A
16. C

122 E
 ALLEN
® Vector
Do yourself - 7

r 2 ˆ ˆ
1.
3
(
r = i + j - 2kˆ ) 2. D 3. (a) C; (b) A,D

5. w = –v1 + 4v2 – 2v3

Do yourself - 8

2. -ˆi - ˆj + 2kˆ 4. B 5. C 6. C 7. A

8. (A) T; (B) U ; (C) P; (D) R ; (E) Q; (F) S; (G) W; (H) V 9. C

Do yourself - 9

®
1 1 3 2
1. 2. 3. 4. A 5. B,C,D
6 14 2

Do yourself - 10
1. ±1 2. 0 3. 7 4. Right handed system
5. D 6. C 7. C 8. C

Do yourself - 11
1. B 2. B 3. C 4. B

Do yourself - 12

1. 62, 92iˆ + 102ˆj + 32kˆ 2. A 3. B 4. C

5. A,B,D 6. B,D
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

Do yourself - 13
1. 3 2. linearly dependent. 3. D 4. D
5. D 6. D 7. B,C,D

Do yourself - 14
1. B 2. 4 3. 1 4. B 5. A
6. B 7. C 8. 9

E 123
JEE-Mathematics ALLEN
®

EXERCISE # O-1

Que. 1 2 3 4 5 6 7 8 9 10
Ans. D B B C A (i) D, (ii) B, (iii) B C B D A
Que. 11 12 13 14 15 16 17 18 19 20
Ans. A A A C B A C C D D
Que. 21 22 23 24 25 26 27 28 29 30
Ans. C B A D C A D A C A

EXERCISE # O-2

®
Que. 1 2 3 4 5 6 7 8 9 10
Ans. A,C A,B,D B,C A,C,D A,C,D B,C C,D A,B,D A,B,C,D A,D
Que. 11 12 13 14 15 16 17 18 19 20
Ans. A,C A,B,D B,C,D A,B,C,D A,B,C B,D A,B,C A,C B,D A,B,C,D
Que. 21 22 23 24 25
Ans. B,D A,B A,C B,D A,B,D

EXERCISE # O-3

Que. 1 2 3 4 5 6 7 8 9
Ans. C B C B A 2.41 or 2.42 2.44 or 2.45 0.00 B
A B C D
Q.10
R Q,S Q,R P,Q,R,S
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

EXERCISE # O-4

Que. 1 2 3 4 5 6 7 8 9 10
Ans. 3 0 5 3 7 3.80 1.04 or 1.05 0.45 or 0.46 5.00 4.00

124 E
 ALLEN
® Vector

EXERCISE # JEE-MAIN

Que. 1 2 3 4 5 6 7 8 9 10
Ans. 3 1 1 4 3 1 3 4 1 2
Que. 11 12 13 14 15 16 17 18 19 20
Ans. 3 1 1 3 1 30 1.00 8.00 2 4
Que. 21 22 23 24
Ans. 2 36 C A

®
EXERCISE # JEE-ADVANCED

Que. 1 2 3 4 5 6 7 8 9 10
Ans. C 5 C A,B,C 4 A,C,D Bonus B,C B D
Que. 11 12 13 14 15 16 17 18 19
Ans. B 3 3,4 18.00 108.00 A,C 7 A,B,C B,C,D
node06\B0BC-BD\Kota\JEE(Advanced)\Enthuse\Maths\Module\4-Module#Matrices, Vector & 3D\Eng\06-Vector_(Exe)_Part-2.p65

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