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

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.2.

Board	CBSE
Textbook	NCERT
Class	Class 12
Subject	Maths
Chapter	Chapter 10
Chapter Name	Vector Algebra
Exercise	Ex 10.2
Number of Questions Solved	19
Category	NCERT Solutions

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

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:
$\overrightarrow { a } =\hat { i } +\hat { j } +\hat { k }$
$\left| \overrightarrow { a } \right| =\sqrt { { 1 }^{ 2 }+{ 1 }^{ 2 }+{ 1 }^{ 2 } }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 1

Question 2.
Write two different vectors having same magnitude.
Solution:
$\overrightarrow { a } =\hat { i } +\hat { 2j } +\hat { 3k } ,\overrightarrow { b } =\hat { 3i } +\hat { 2j } +\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 2
Such possible answers are infinite

Question 3.
Write two different vectors having same direction.
Solution:
Let the two vectors be
$\overrightarrow { a } =\hat { i } +\hat { j } +\hat { k } ,\overrightarrow { b } =\hat { 3i } +\hat { 3j } +\hat { 3k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 3
Hence vectors $\overrightarrow { a } ,\overrightarrow { b }$ have the same direction but different magnitude

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

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 $-\hat { 7i } and\hat { 6j }$ and scalar components are – 7 and 6.

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

Question 7.
Find the unit vector in the direction of the vector
$\overrightarrow { a } =\hat { i } +\hat { j } +\hat { 2k }$
Solution:
$\overrightarrow { a } =\hat { i } +\hat { j } +\hat { 2k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 7

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
$\overrightarrow { PQ } =(4-1)\hat { i } +(5-2)\hat { j } +(6-3)\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 8

Question 9.
For given vectors $\overrightarrow { a } =2\hat { i } -\hat { j } +2\hat { k } \quad and\quad \overrightarrow { b } =-\hat { i } +\hat { j } -\hat { k }$ find the unit vector in the direction of the vector $\overrightarrow { a } +\overrightarrow { b }$
Solution:
$\overrightarrow { a } =2\hat { i } -\hat { j } +2\hat { k } \quad and\quad \overrightarrow { b } =-\hat { i } +\hat { j } -\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 9

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

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

Question 12.
Find the direction cosines of the vector $\hat { i } +2\hat { j } +3\hat { k }$
Solution:
let $\overrightarrow { p } =\hat { i } +2\hat { j } +3\hat { k }$
Now a = 1,b = 2,c = 3
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 12

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
$({ x }_{ 2 }-{ x }_{ 1 })\hat { i } +({ y }_{ 2 }-{ y }_{ 1 })\hat { j } +({ z }_{ 2 }-{ z }_{ 1 })\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 13

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 $x\hat { i } +y\hat { j } +z\hat { k }$ are
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 14
which shows that the vector a is equally inclined to the axes OX, OY, OZ.

Question 15.
Find the position vector of a point R which divides the line joining the points whose positive vector are $P(\hat { i } +2\hat { j } -\hat { k } )\quad and\quad Q(-\hat { i } +\hat { j } +\hat { k } )$ in the ratio 2:1
(i) internally
(ii) externally.
Solution:
(i) The point R which divides the line joining the point $P(\overrightarrow { a } )\quad and\quad Q(\overrightarrow { b } )$ in the ratio m : n
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 15

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 $\overrightarrow { OP } =2\hat { i } +3\hat { j } +4\hat { k } \quad and\quad \overrightarrow { OQ } =4\hat { i } +\hat { j } -2\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 16

Question 17.
Show that the points A, B and C with position vector $\overrightarrow { a } =3\hat { i } -4\hat { j } -4\hat { k } ,\overrightarrow { b } =2\hat { i } -\hat { j } +\hat { k } and\quad \overrightarrow { c } =\hat { i } -3\hat { j } -5\hat { k }$ respectively form the vertices of a right angled triangle.
Solution:
$\overrightarrow { AB } =\overrightarrow { b } -\overrightarrow { a } =-\hat { i } +3\hat { j } +5\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 17

Question 18.
In triangle ABC (fig.), which of the following is not
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 18
(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

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

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

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