# NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3

In this chapter, we provide NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 for English medium students, Which will very helpful for every student in their exams. Students can download the latest NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 pdf, free NCERT solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 book pdf download.

## NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3

Ex 10.3 Class 12 Maths Question 1.
Find the angle between two vectors $overrightarrow { a } ,overrightarrow { b }$ with magnitudes √3 and 2 respectively, and such that $overrightarrow { a } cdot overrightarrow { b } =sqrt { 6 }$
Solution:
Angle θ between two vectors $overrightarrow { a } ,overrightarrow { b }$,

Ex 10.3 Class 12 Maths Question 2.
Find the angle between the vectors $hat { i } -2hat { j } +3hat { k } quad andquad 3hat { i } -2hat { j } +hat { k }$
Solution:
Let $overrightarrow { a } =hat { i } -2hat { j } +3hat { k } quad andquad overrightarrow { b } =3hat { i } -2hat { j } +hat { k }$
Let θ be the angle between $overrightarrow { a } ,overrightarrow { b }$,

Ex 10.3 Class 12 Maths Question 3.
Find the projection of the vector $overrightarrow { i } -overrightarrow { j }$, on the line represented by the vector $overrightarrow { i } +overrightarrow { j }$,
Solution:
let $overrightarrow { a } =hat { i } -hat { j } quad andquad overrightarrow { b } =hat { i } +hat { j }$

Ex 10.3 Class 12 Maths Question 4.
Find the projection of the vector $hat { i } +3hat { j } +7hat { k }$ on the vector $7hat { i } -hat { j } +8hat { k }$
Solution:
let $overrightarrow { a } =hat { i } +3hat { j } +7hat { k } quad andquad overrightarrow { b } =7hat { i } -hat { j } +8hat { k }$ then

Ex 10.3 Class 12 Maths Question 5.
Show that each of the given three vectors is a unit vector $frac { 1 }{ 7 } left( 2hat { i } +3hat { j } +6hat { k } right) ,frac { 1 }{ 7 } left( 3hat { i } -6hat { j } +2hat { k } right) ,frac { 1 }{ 7 } left( 6hat { i } +2hat { j } -3hat { k } right)$ Also show that they are mutually perpendicular to each other.
Solution:
$Letquad overrightarrow { a } =frac { 1 }{ 7 } left( 2hat { i } +3hat { j } +6hat { k } right) ,overrightarrow { b } =frac { 1 }{ 7 } left( 3hat { i } -6hat { j } +2hat { k } right) ,overrightarrow { c } =frac { 1 }{ 7 } left( 6hat { i } +2hat { j } -3hat { k } right)$

Ex 10.3 Class 12 Maths Question 6.
$Findleft| overrightarrow { a } right| andleft| overrightarrow { b } right| ifleft( overrightarrow { a } +overrightarrow { b } right) cdot left( overrightarrow { a } -overrightarrow { b } right) =8quad andleft| overrightarrow { a } right| =8left| overrightarrow { b } right|$
Solution:
Given $left( overrightarrow { a } +overrightarrow { b } right) cdot left( overrightarrow { a } -overrightarrow { b } right) =8$

Ex 10.3 Class 12 Maths Question 7.
Evaluate the product :
$left( 3overrightarrow { a } -5overrightarrow { b } right) cdot left( 2overrightarrow { a } +7overrightarrow { b } right)$
Solution:
$left( 3overrightarrow { a } -5overrightarrow { b } right) cdot left( 2overrightarrow { a } +7overrightarrow { b } right)$
$=6overrightarrow { a } .overrightarrow { a } -10overrightarrow { b } overrightarrow { a } +21overrightarrow { a } .overrightarrow { b } -35overrightarrow { b } .overrightarrow { b }$
$=6{ left| overrightarrow { a } right| }^{ 2 }-11overrightarrow { a } overrightarrow { b } -35{ left| overrightarrow { b } right| }^{ 2 }$

Ex 10.3 Class 12 Maths Question 8.
Find the magnitude of two vectors $overrightarrow { a } ,overrightarrow { b }$ having the same magnitude and such that the angle between them is 60° and their scalar product is $frac { 1 }{ 2 }$
Solution:
We know that $overrightarrow { a } .overrightarrow { b } =left| overrightarrow { a } right| left| overrightarrow { b } right| costheta$

Ex 10.3 Class 12 Maths Question 9.
Find $left| overrightarrow { x } right|$ , if for a unit vector $overrightarrow { a } ,(overrightarrow { x } -overrightarrow { a } )cdot (overrightarrow { x } +overrightarrow { a } )=12$
Solution:
Given
$overrightarrow { a } ,(overrightarrow { x } -overrightarrow { a } )cdot (overrightarrow { x } +overrightarrow { a } )=12$

Ex 10.3 Class 12 Maths Question 10.
If $overrightarrow { a } =2hat { i } +2hat { j } +3hat { k } ,overrightarrow { b } =-hat { i } +2hat { j } +hat { k } andoverrightarrow { c } =3hat { i } +hat { j }$ such that $overrightarrow { a } +lambda overrightarrow { b } bot overrightarrow { c }$ , then find the value of λ.
Solution:
Given
$overrightarrow { a } =2hat { i } +2hat { j } +3hat { k } ,overrightarrow { b } =-hat { i } +2hat { j } +hat { k } andoverrightarrow { c } =3hat { i } +hat { j }$

Ex 10.3 Class 12 Maths Question 11.
Show that $left| overrightarrow { a } right| overrightarrow { b } +left| overrightarrow { b } right| aquad bot quad left| overrightarrow { a } right| cdot overrightarrow { b } -left| overrightarrow { b } right| a$ for any two non-zero vectors $overrightarrow { a } ,overrightarrow { b }$
Solution:
$overrightarrow { a } ,overrightarrow { b }$ are any two non zero vectors

Ex 10.3 Class 12 Maths Question 12.
If $overrightarrow { a } cdot overrightarrow { a } =0quad andquad overrightarrow { a } cdot overrightarrow { b } =0$, then what can be concluded about the vector $overrightarrow { b }$ ?
Solution:
$overrightarrow { a } overrightarrow { a } =0quad andquad overrightarrow { a } .overrightarrow { b } =0$,
=> $overrightarrow { b }$ = 0
Hence b is any vector.

Ex 10.3 Class 12 Maths Question 13.
If $overrightarrow { a } ,overrightarrow { b } ,overrightarrow { c }$ are the unit vector such that $overrightarrow { a } +overrightarrow { b } +overrightarrow { c } =0$ , then find the value of $overrightarrow { a } .overrightarrow { b } +overrightarrow { b } .overrightarrow { c } +overrightarrow { c } .overrightarrow { a }$
Solution:
We have
$overrightarrow { a } +overrightarrow { b } +overrightarrow { c } =0$

Ex 10.3 Class 12 Maths Question 14.
If either vector $overrightarrow { a } =0quad orquad overrightarrow { b } =0$ then $overrightarrow { a } .overrightarrow { b } =0$. But the converse need not be true. Justify your answer with an example.
Solution:
Given: $overrightarrow { a } =0quad orquad overrightarrow { b } =0$
To prove: $overrightarrow { a } .overrightarrow { b } =0$

Ex 10.3 Class 12 Maths Question 15.
If the vertices A,B,C of a triangle ABC are (1,2,3) (-1,0,0), (0,1,2) respectively, then find ∠ABC.
Solution:
Let O be the origin then.
$frac { 1 }{ 2 }$

Ex 10.3 Class 12 Maths Question 16.
Show that the points A (1,2,7), B (2,6,3) and C (3,10, -1) are collinear.
Solution:
The position vectors of points A, B, C are

Ex 10.3 Class 12 Maths Question 17.
Show that the vectors $2hat { i } -hat { j } +hat { k } ,hat { i } -3hat { j } -5hat { k }$ and $left( 3hat { i } -4hat { j } -4hat { k } right)$ from the vertices of a right angled triangle.
Solution:
The position vectors of the points A, B and C are
$2hat { i } -hat { j } +hat { k } ,hat { i } -3hat { j } -5hat { k }$ and $left( 3hat { i } -4hat { j } -4hat { k } right)$

Ex 10.3 Class 12 Maths Question 18.
If $overrightarrow { a }$ is a non-zero vector of magnitude ‘a’ and λ is a non- zero scalar, then λ $overrightarrow { a }$ is unit vector if
(a) λ = 1
(b) λ = – 1
(c) a = |λ|
(d) a = $frac { 1 }{ left| lambda right| }$
Solution:
$left| overrightarrow { a } right| =a$
Given : $lambda overrightarrow { a }$ is a unit vectors

All Chapter NCERT Solutions For Class12 Maths

—————————————————————————–

All Subject NCERT Solutions For Class12

*************************************************

Remark:

I think you got complete solutions for this chapter. If You have any queries regarding this chapter, please comment on the below section our subject teacher will answer you. We tried our best to give complete solutions so you got good marks in your exam.

If these solutions have helped you, you can also share Careerkundali.in to your friends.

Best of Luck!!