SRI SRI RAVISHANKAR VIDYA MANDIR , INDORE

SUBJECT - MATHEMATICS
DATE: 12/08/2023 WORKSHEET:- VRCTORS AND 3-D CLASS – XII

1
Find the unit vector in the direction of the sum of the vectors a = 2 iˆ  ˆj  2 kˆ and b = – iˆ 
ˆj  3 k ˆ .
2
Find a vector of magnitude 11 in the direction opposite to that of PQ , where P and Q are the
points (1, 3, 2) and (–1, 0, 8), respectively.
3
If the points (–1, –1, 2), (2, m, 5) and (3,11, 6) are collinear, find the value of m.
4
Find all vectors of magnitude 10√ 3of iˆ 2 ˆj  kˆ and iˆ 3 ˆj  4kˆ .
that are perpendicular to the plane
5
The value of  for which the two vectors 2iˆ ˆj  2kˆ and 3iˆ ˆj  kˆ are perpendicular .
6

The area of the parallelogram whose adjacent sides are 2iˆ ˆj  kˆ and
 iˆ kˆ is= ……

7 The 2 vectors ˆj  kˆ and 3iˆ ˆj  4kˆ represents the two sides AB and AC, respectively of a
∆ABC. The length of the median through A is=……
8 The unit vector perpendicular to the vectors iˆ  ˆj and iˆ  ˆj forming a right handed system is
9 A vector r is inclined at equal angles to the three axes. If the magnitude of r is 2√ 3 units , find r
1 A vector r has magnitude 14 and direction ratios 2, 3, – 6. Find the direction cosines and
0 components of r , given that r makes an acute angle with x-axis.
1 Find a vector of magnitude 6, which is perpendicular to both the vectors 2iˆ  ˆj 2kˆ and 4iˆ – ˆj 
1
3kˆ .
1 Find the sine of the angle between the vectors a  3i  j  2k and b  2iˆ  2 ˆj  4kˆ
2
1 If A, B, C, D are the points with position vectors iˆ ˆj  kˆ , 2iˆ ˆj  3kˆ , 2iˆ 3kˆ , 3iˆ
3
2 ˆ j  kˆ
respectively, find the projection of AB along CD .
1 Using vectors, find the area of the triangle ABC with vertices A(1, 2, 3), B(2, – 1, 4) and C(4,
4 5, – 1).

1 a , b⃗ , c⃗ are mutually perpendicular vectors of equal magnitudes, show that the vector c⃗ .d⃗ =15
If ⃗
5 is equally inclined to⃗ ⃗ c .
a , b∧⃗
1 The scalar product of the vector iˆ ˆj + kˆ with a unit vector along the sum of vectors 2iˆ 4 ˆj 
6
5kˆ and λiˆ 2 ˆj+3 kˆ is equal to one. Find the value of λ
1 IF a  1i  4j  2k , b  3iˆ  2 ˆj 7kˆ and c⃗ = 2iˆ ˆj  4kˆ.Find a vector which is perpendicular
7 to both ⃗a ,∧b⃗ andc⃗ .d⃗ =15 .
1 Show that the points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7) are collinear, and find the ratio in
8 which B divides AC.
1 Find the value of x for which x (iˆ ˆj + kˆ ) is a unit vector
9
2 A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops.
0 Determine the girl’s displacement from her initial point of departure.
2 Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-
1 axis.
2 Write all the unit vectors in XY-plane
2
THREE DIMENSIONAL GEOMETRY

2
If the direction ratios of a line are 1, 1, 2, find the direction cosines of the line.
3
2
Find the direction cosines of the line passing through the points P (2, 3, 5) and Q (–1, 2, 4).
4
2
5 The x-coordinate of a point on the line joining the points Q (2, 2, 1) and R (5, 1, –2) is 4. Find its
z-coordinate.
2 Find the co-ordinates of the foot of perpendicular drawn from the point A (1, 8, 4) to the line
6 joining the points B (0, –1, 3) and C (2, –3, –1).
2 Prove that the line through A (0, –1, –1) and B (4, 5, 1) intersects the line through C (3, 9, 4) and
D (– 4, 4, 4).
7
2 x y−1 z−1
Find the image of the point (1, 6, 3) in the line = = .
8 1 2 3
2 x−3 y−3 z
Find the equations of the two lines through the origin which intersect the line = = .
9 2 1 1
at angles of π /3each.
3 Find the angle between the lines whose direction cosines are given by the equations l + m + n =
0 0, l2 + m2 – n 2 = 0.
3 4−x y 1−z
Find the foot of perpendicular from the point (2,3,–8) to the line = = . Also, find the
1 2 6 3
perpendicular distance from the given point to the line.
3 x+5 y+ 3 z−6
Find the distance of a point (2,4,–1) from the line = = .
2 1 4 −9
→
3 Find the shortest distance between the lines given r⃗ (8  3iˆ (9  16) ˆj +(10  7)k and r⃗
3 15iˆ 29 ˆj  5 kˆ (3iˆ 8 ˆj  5kˆ) .
3
4 Show that the straight lines whose direction cosines are given by 2l + 2m – n = 0 and mn + nl +
lm = 0 are at right angles.
3 x−1 y−2 z−3 x−4 y−1 z
Show that the line = = and = = intersect .Find points of intersection.
5 2 3 4 2 3 1
3 x−1 y+ 1 z−1 x+2 y−1 z+1
Show that the line = = and = = do not intersect .
6 3 2 5 4 3 −2
3 ^ ^j− k^ ) + (3^i− ^j ) and r⃗ = (4 i−
Show that the lines r⃗ = (i+ ^ k^ ) + (2^i+3 k^ ) intersect . Find points
7 of intersection.
3 x+3 y−1 z+ 4
Find the foot of the perpendicular from the point (0,2,3) on the line = = . Also
8 5 2 3
find image of (0,2, 3) and length of the perpendicular .
3 The equation of a line, which is parallel to 2iˆ  ˆj  3kˆ and which passes through the point (5,–
9 x−5 y+ 2 z−4
2,4), is = = is correct or wrong.
2 −1 3
4 → →
Find the shortest distance between the lines r⃗ (iˆ+() ˆj -(1  )kˆ and r⃗ (iˆ+
0
() ˆj +(  kˆ
4 Determine the vector equation of a line passing through (1, 2, –4) and perpendicular to the two
1 lines
r⃗ 8iˆ - 16 ˆj  10 kˆ  (3iˆ ˆj +7kˆ) and r⃗ iˆ +29 ˆj  5kˆ  (3iˆ8 ˆj -5kˆ).
4 Find the equation of a line parallel to x-axis and passing through the origin.
2
4 If the coordinates of the points A, B, C, D be (1, 2, 3), (4, 5, 7), (– 4, 3, – 6) and (2, 9, 2)
3 respectively, then find the angle between the lines AB and CD.
4 x−1 y +1 z−1 x+2 1− y z+1
If the lines = = and = = are perpendicular, find the value of k.
4 3 −2 k 5 3k −1 −5
4 Find the vector equation of the line passing through the point (1, 2, – 4) and perpendicular to
5 x−8 y+ 19 z−10 x−15 y−29 z−5
the two lines: = = and = =
3 −16 7 3 8 −5
ARUN DANGI (PGT MATHEMATICS)