# NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2

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## NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2

Ex 10.2 Class 12 Maths Question 1.
Compute the magnitude of the following vectors:
$overrightarrow { a } =hat { i } +hat { j } +hat { k } ,overrightarrow { b } =hat { 2i } -hat { 7j } -hat { 3k }$
$overrightarrow { c } =frac { 1 }{ sqrt { 3 } } hat { i } +frac { 1 }{ sqrt { 3 } } hat { j } -frac { 1 }{ sqrt { 3 } } hat { k }$
Solution:
$overrightarrow { a } =hat { i } +hat { j } +hat { k }$
$left| overrightarrow { a } right| =sqrt { { 1 }^{ 2 }+{ 1 }^{ 2 }+{ 1 }^{ 2 } }$

Ex 10.2 Class 12 Maths Question 2.
Write two different vectors having same magnitude.
Solution:
$overrightarrow { a } =hat { i } +hat { 2j } +hat { 3k } ,overrightarrow { b } =hat { 3i } +hat { 2j } +hat { k }$

Ex 10.2 Class 12 Maths Question 3.
Write two different vectors having same direction.
Solution:
Let the two vectors be
$overrightarrow { a } =hat { i } +hat { j } +hat { k } ,overrightarrow { b } =hat { 3i } +hat { 3j } +hat { 3k }$

Hence vectors $overrightarrow { a } ,overrightarrow { b }$ have the same direction but different magnitude

Ex 10.2 Class 12 Maths Question 4.
Find the values of x and y so that the vectors $overrightarrow { 2i } +overrightarrow { 3j } quad andquad hat { xi } +hat { yj }$ are equal.
Solution:
We are given $overrightarrow { 2i } +overrightarrow { 3j } quad andquad hat { xi } +hat { yj }$
If vectors are equal, then their respective components are equal. Hence x = 2, y = 3.

Ex 10.2 Class 12 Maths Question 5.
Find the scalar and vector components of the vector with initial point (2,1) and terminal point (-5,7).
Solution:
LetA(2, 1) be the initial point and B(-5,7) be the terminal point $overrightarrow { AB } =left( { x }_{ 2 }-{ x }_{ 1 } right) hat { i } +left( { y }_{ 2 }-{ y }_{ 1 } right) hat { j } =-hat { 7i } +hat { 6j }$
∴The vector components are $-hat { 7i } andhat { 6j }$ and scalar components are – 7 and 6.

Ex 10.2 Class 12 Maths Question 6.
Find the sum of three vectors:
$overrightarrow { a } =hat { i } -hat { 2j } +hat { k } ,overrightarrow { b } =-2hat { i } +hat { 4j } +5hat { k } quad andquad overrightarrow { c } =hat { i } -hat { 6j } -hat { 7k } ,$
Solution:
$overrightarrow { a } =hat { i } -hat { 2j } +hat { k } ,overrightarrow { b } =-2hat { i } +hat { 4j } +5hat { k } quad andquad overrightarrow { c } =hat { i } -hat { 6j } -hat { 7k } ,$
$overrightarrow { a } +overrightarrow { b } +overrightarrow { c } =hat { 0i } -hat { 4j } -hat { k } =-4hat { i } -hat { k }$

Ex 10.2 Class 12 Maths Question 7.
Find the unit vector in the direction of the vector
$overrightarrow { a } =hat { i } +hat { j } +hat { 2k }$
Solution:
$overrightarrow { a } =hat { i } +hat { j } +hat { 2k }$

Ex 10.2 Class 12 Maths Question 8.
Find the unit vector in the direction of vector $overrightarrow { PQ }$, where P and Q are the points (1,2,3) and (4,5,6) respectively.
Solution:
The points P and Q are (1, 2, 3) and (4, 5, 6) respectively
$overrightarrow { PQ } =(4-1)hat { i } +(5-2)hat { j } +(6-3)hat { k }$

Ex 10.2 Class 12 Maths Question 9.
For given vectors $overrightarrow { a } =2hat { i } -hat { j } +2hat { k } quad andquad overrightarrow { b } =-hat { i } +hat { j } -hat { k }$ find the unit vector in the direction of the vector $overrightarrow { a } +overrightarrow { b }$
Solution:
$overrightarrow { a } =2hat { i } -hat { j } +2hat { k } quad andquad overrightarrow { b } =-hat { i } +hat { j } -hat { k }$

Ex 10.2 Class 12 Maths Question 10.
Find a vector in the direction of $5hat { i } -hat { j } +2hat { k }$ which has magnitude 8 units.
Solution:
The given vector is $overrightarrow { a } =5hat { i } -hat { j } +2hat { k }$

Ex 10.2 Class 12 Maths Question 11.
Show that the vector $2hat { i } -3hat { j } +4hat { k } quad andquad -4hat { i } +6hat { j } -8hat { k }$ are collinear.
Solution:
$overrightarrow { a } =2hat { i } -3hat { j } +4hat { k } quad andquad overrightarrow { b } =-4hat { i } +6hat { j } -8hat { k }$
$=-2(2hat { i } -3hat { j } +4hat { k } )$
vector $overrightarrow { a } quad andquad overrightarrow { b }$ have the same direction they are collinear.

Ex 10.2 Class 12 Maths Question 12.
Find the direction cosines of the vector $hat { i } +2hat { j } +3hat { k }$
Solution:
let $overrightarrow { p } =hat { i } +2hat { j } +3hat { k }$
Now a = 1,b = 2,c = 3

Ex 10.2 Class 12 Maths Question 13.
Find the direction cosines of the vector joining the points A (1,2, -3) and B(-1, -2,1), directed fromAtoB.
Solution:
Vector joining the points A and B is
$({ x }_{ 2 }-{ x }_{ 1 })hat { i } +({ y }_{ 2 }-{ y }_{ 1 })hat { j } +({ z }_{ 2 }-{ z }_{ 1 })hat { k }$

Ex 10.2 Class 12 Maths Question 14.
Show that the vector $hat { i } +hat { j } +hat { k }$ are equally inclined to the axes OX, OY, OZ.
Solution:
Let $hat { i } +hat { j } +hat { k } =overrightarrow { a }$ , Direction cosines of vector $xhat { i } +yhat { j } +zhat { k }$ are

which shows that the vector a is equally inclined to the axes OX, OY, OZ.

Ex 10.2 Class 12 Maths Question 15.
Find the position vector of a point R which divides the line joining the points whose positive vector are $P(hat { i } +2hat { j } -hat { k } )quad andquad Q(-hat { i } +hat { j } +hat { k } )$ in the ratio 2:1
(i) internally
(ii) externally.
Solution:
(i) The point R which divides the line joining the point $P(overrightarrow { a } )quad andquad Q(overrightarrow { b } )$ in the ratio m : n

Ex 10.2 Class 12 Maths Question 16.
Find position vector of the mid point of the vector joining the points P (2,3,4) and Q (4,1, -2).
Solution:
Let $overrightarrow { OP } =2hat { i } +3hat { j } +4hat { k } quad andquad overrightarrow { OQ } =4hat { i } +hat { j } -2hat { k }$

Ex 10.2 Class 12 Maths Question 17.
Show that the points A, B and C with position vector $overrightarrow { a } =3hat { i } -4hat { j } -4hat { k } ,overrightarrow { b } =2hat { i } -hat { j } +hat { k } andquad overrightarrow { c } =hat { i } -3hat { j } -5hat { k }$ respectively form the vertices of a right angled triangle.
Solution:
$overrightarrow { AB } =overrightarrow { b } -overrightarrow { a } =-hat { i } +3hat { j } +5hat { k }$

Ex 10.2 Class 12 Maths Question 18.
In triangle ABC (fig.), which of the following is not

(a) $overrightarrow { AB } +overrightarrow { BC } +overrightarrow { CA } =overrightarrow { 0 }$
(b) $overrightarrow { AB } +overrightarrow { BC } -overrightarrow { AC } =overrightarrow { 0 }$
(c) $overrightarrow { AB } +overrightarrow { BC } -overrightarrow { CA } =overrightarrow { 0 }$
(d) $overrightarrow { AB } -overrightarrow { CB } +overrightarrow { CA } =overrightarrow { 0 }$
Solution:
We know that
$overrightarrow { AB } +overrightarrow { BC } +overrightarrow { CA } =overrightarrow { 0 }$
$overrightarrow { AB } +overrightarrow { BC } -overrightarrow { AC } =overrightarrow { 0 }$
Hence option (c) is not correct

Ex 10.2 Class 12 Maths Question 19.
If $overrightarrow { a } ,overrightarrow { b }$ are two collinear vectors then which of the following are incorrect:
(a) $overrightarrow { b } =lambda overrightarrow { a }$, for some scalar λ.
(b) $overrightarrow { a } =pm overrightarrow { b }$
(c) the respective components of $overrightarrow { a } ,overrightarrow { b }$ are proportional.
(d) both the vectors $overrightarrow { a } ,overrightarrow { b }$ have same direction, but different magnitudes.
Solution:
Options (d) is incorrect since both the vectors $overrightarrow { a } ,overrightarrow { b }$ , being collinear, are not necessarily in the same direction. They may have opposite directions. Their magnitudes may be different.

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