roblems For Practice

A=3+2j and B=i -2j + 3k, find the

magnitudes of A + B and A -

B. (Ans. 5, V29)
resultant of the
Find the unit vector parallel to the A A
A

vectors A =2i -

6j -

3k and B =4i +3j -k.

LAns. (6i-3j - 4 J

added to the resul-

Determine the vector which when
A

6k and B = 4i +3j + 3k gives

tantof A = 2i -

4j -

the unit vector along z-axis. (Ans. 6 i +j+4k)

unit vector
the value of in the
4 Find
A
(Ans. V0.2)
^k.
0.4 i+0.8 j +
A A

vectors a = 4i -j,
Given three coplanar
A

3i +2j andc =-3j. Find the magnitude of

b = -

vectors.
(Ans. 5)
the sum of the three
the horizontal. If its
inclined at 30° to
6. A force is direction
in the horizontal
rectangular component the force and its
the magnitude of
is 50 N, find 57.74 N, 28.87 N)
vertical component.
(Ans.
ms' has its Y-component 5/2 ms
velocity of 10
IA (Ans. 5V2 ms)
Calculate its X-component.
at a n angle of
30° to the
takes off
8. An aeroplane of its velocity along the
horizontal. If the component
what is its actual velocity ?
kmh,
horizontal is 200
component of its velocity.
the vertical
Also find
230.94 km h , 115.47 km hl)
(Ans.
or roblems For Practice

Find the angle between the vectors

A =2i-4j+ 6k and B =3i+j+26.

(Ans. 60
Find the angles between the vectors A = i + i+k

and B --i-j+ 2k. Delhi 951 (Ans. 90

what value of m, is the vector
For
A =2i+3j -6k perpendicular to the vector

B -31-m+6k ? Delhi 161 (Ans. -10)

4For of a
what value are the vectors A =
a
i -2 +k
-4k perpendicular to each
and B =20 i +aj
other? (Ans. 2,-
Find the angles between the following pairs of
vectors

e A-i+j+k and B =-2i-21-2k.

(Ans. 180)

Ain A=-2i+2j-k and B=3i +6j +2

(Ans. 79°

j A=4+6j -3 and B--2i-5j +7k

(Ans. 148.9°)

Calculate the values of () j.(2i -3j+k )and

(i)

(2-).(3+ [Ans. (i) -3(i) 6]

A force F =4i +j + 3k newton acts on a particle
and displaces it through displacement
metre. Calculate the work
S 11+ 11j+ 15k
done by the force.
(Ans. 100 )

8. Under a force of 10i-3j + 6k newton, a body of

mass 5 kg is displaced from the position

6+5-3 to the position 10-2+7k.

(Ans. 121 )
Calculate the work done.
The sum and difference of two vectors A and B are

A+ B-2+ 6 +k and A- B-4i+2j-11k.

Find the magnitude of each vector and their scalar

product A. B. (Ans. 50, V41,

-

25)

. A force F =5i+ 4j newton displaces a body

through $ = 3í+ 4k metre in 3s. Find the power.

(Ans. 5 W)
 If the resultant of the
vectors 3i +4 + 5k and
5i+3 + 4k makes
find cos 0.
an
angle with X-axis, then
(Ans. 0.5744)
Show that the vectors
A-3-2+,
B=i-3+ 5 and C -2i+i-4k form right a
,angle triangle.
If vectors A, B and C
have magnitudes 8, 15 and 17
units and A + B =
C, find the angle between A and
B.
(Ans. 90)
14.) If A =B C, then determine the angle between A
-

and B. B2 -(2
0= COs - 1 A
+
Ans.
2 AB

For two vectors A and B if Á+ B= C and

A+ B= C, then prove that A and B are parallel to
each other.

Prove that
(A + 2 B). (2A -

3 B ) 2A2
= + ABcos -6B.
 roblems for Practice
If A=i+3+2 and B =3í+j+2k, then find
the vector product Áx B.
(Ans. 4i+4j -8k)
Prove that the vectors Å =4î+3f +k and
B 12i+9j +3k are parallel to each other.

If A=2 +3 + and -3f+2/+4k, then

find the value of (A + B) x (Á - B)

(Ans. 20i+ 10 + 10 k)
Find the value of a for which the vectors
3f+3+9k andi +aj +3k are parallel.
(Ans. a = 1)

unit vector perpendicular to the vectors

5ind a

A-4f-+3k and B=-2i+j-2k.

Ans. -i+2+2fn.
O 6Find
A=3i-4j+5
the sine of the angle
and B=i-i+k.
between the vectors

(Ans. 1/5)
18 which is
Find a vector of magnitude
4i-j+3k and
to both the vectors
perpendicular

-2f+-2k. (Ans. -6i+ 12j + 12k)

whose
the area of the parallelogram
&Determine the vectors
sides are formed by
adjacent
A-f-3f+k and B=i+j+k.
(Ans. 4/2 square units)
formed by points O, A
of the triangle
Find the area

and
and B such that OA =i+2j+3k
OB--3-2j+k. (Ans. 3/5 square units)
area of the
the help of vectors, the
Find with
10.
with vertices
1, 2) B(1, -1, -3) and
A(3,
triangle (Ans. V165/2 sq. units)
C(4,-3, 1).
vectors that|A|=2
two such
11. If A and B are

B =3i+2j+6k,findthe
angle
B=7 and A x

(Ans. t/ 6)
between A and B.
of
point i'+ 2j-k
a

moment
about the
12. Find the the
acting through
force represented
by 3i +k
3 f + 11f+ 9£)
point 2 i-j+ 3k. (Ans.
Prove that ( +) x(a
-8)=2(B xa).
14Prove that | a xb| =ya- (a.b
Find | A x B| if| Á | 10,|
=
B| 2 and A. B =12
=

(Ans. 16)
. a
=i-2-3k, b =2--k and
C =î+3 i -2k, find (ax b) xc.

(Ans.f++2)
17Calculate the area of the triangle determined by the
A
two vectors: A =3i +4j and B =-3i+7.
ICentral Schools 08, 09]
(Ans. 16.5 square units)
18 Find the area of a
parallelogram formed by vectors
A 31 +2, B= -3í+7
=

Central Schools 071

(Ans. 33 square units)
The diagonals of a parallelogram are represented

by R -3i+2j-7k and R =5i+6-3X. Find

the area of the parallelogram.
IChandigarh 071
(Ans. 2v509 square units)
 between a 2N force and
Calculate the angle
force sothat their resultant is 4N. (Ans. 75
da3N

resultant vector of P and Q is R.On reua

The ing
the direction of Q, the resultant vector becomes S.
S =2(p + O).
Showthat:R +

have the square of theirresultans

Two equal forces
to three times
their product. Find the angle
equal
between them. (Ans. 60)
betweenof equal
two vectorsS
When the angle
is 21/3 prove that the magnitude of the
magnitude
resultantis equal to either.
the two forces (P+Q)and (P- )
At what angle do
9 actso that the resultant is 3P+ Q. (Ans. 60)
[Himachal 041

The resultant of two equal forces acting at right

10. Find the
to each other is 1414 dyne.
angles IHimachal 04
magnitude of either force.
(Ans.1000 dyne)
is acted upon by four forces simul
A particle
taneously:
(i) 30 N due east

(i) 20 N due north

(in) 50 N due west and

the
resultant force on
(iv) 40 N due south. Find the
(Ans.202 N, 45° south of west)
particle. to each
Two boys raising a load pull at an angle
N
N and 60
other. If they exert forces of 30
at rignt
and their effective pull is
respectively
angles to the direction of the pull of the first boy,
? What is t
What is the angle between their arms
303. N)
effective pull? (Ans.120°,
A ship is steaming due east at 12 ms. A woma
uns across the deck at 5 ms-l in a direction at rngnt
and
of the ship
angles to the direction of motion or te
velocity
nen towards north. Calculate the
woman relative to sea.
13 ms, 2237'north
of east)
(Ans.

vectors Pand
Oit
ind
the angle between two

the vectors is given by R2 =P +

r ICentral Schools 071

(Ans. 90°