NCERT Solutions For Class 12 Maths Chapter 10 Vector Algebra Ex 10.3
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Textbook | NCERT |
Board | CBSE |
Category | NCERT Solutions |
Class | Class 12 |
Subject | Maths |
Chapter | Chapter 10 |
Exercise | Class 12 Maths Chapter 10 Vector Algebra Exercise 10.3 |
Number of Questions Solved | 18 |

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3
NCERT TEXTBOOK EXERCISES
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 $\hat { i } -2\hat { j } +3\hat { k } \quad and\quad 3\hat { i } -2\hat { j } +\hat { k }$
Solution:
Let $\overrightarrow { a } =\hat { i } -2\hat { j } +3\hat { k } \quad and\quad \overrightarrow { b } =3\hat { i } -2\hat { j } +\hat { k }$
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 $\overrightarrow { a } =\hat { i } -\hat { j } \quad and\quad \overrightarrow { b } =\hat { i } +\hat { j }$

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

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

Ex 10.3 Class 12 Maths Question 6.
$Find\left| \overrightarrow { a } \right| and\left| \overrightarrow { b } \right| if\left( \overrightarrow { a } +\overrightarrow { b } \right) \cdot \left( \overrightarrow { a } -\overrightarrow { b } \right) =8\quad and\left| \overrightarrow { a } \right| =8\left| \overrightarrow { b } \right|$
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 :
$\left( 3\overrightarrow { a } -5\overrightarrow { b } \right) \cdot \left( 2\overrightarrow { a } +7\overrightarrow { b } \right)$
Solution:
$\left( 3\overrightarrow { a } -5\overrightarrow { b } \right) \cdot \left( 2\overrightarrow { a } +7\overrightarrow { b } \right)$
= $=6\overrightarrow { a } .\overrightarrow { a } -10\overrightarrow { b } \overrightarrow { a } +21\overrightarrow { a } .\overrightarrow { b } -35\overrightarrow { b } .\overrightarrow { b }$
$=6{ \left| \overrightarrow { a } \right| }^{ 2 }-11\overrightarrow { a } \overrightarrow { b } -35{ \left| \overrightarrow { b } \right| }^{ 2 }$
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 1/2
Solution:
We know that $\overrightarrow { a } .\overrightarrow { b } =\left| \overrightarrow { a } \right| \left| \overrightarrow { b } \right| cos\theta$

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 $\overrightarrow { a } =2\hat { i } +2\hat { j } +3\hat { k } ,\overrightarrow { b } =-\hat { i } +2\hat { j } +\hat { k } and\overrightarrow { c } =3\hat { i } +\hat { j }$ such that $\overrightarrow { a } +\lambda \overrightarrow { b } \bot \overrightarrow { c }$ , then find the value of λ.
Solution:
Given
$\overrightarrow { a } =2\hat { i } +2\hat { j } +3\hat { k } ,\overrightarrow { b } =-\hat { i } +2\hat { j } +\hat { k } and\overrightarrow { c } =3\hat { i } +\hat { j }$

Ex 10.3 Class 12 Maths Question 11.
Show that $\left| \overrightarrow { a } \right| \overrightarrow { b } +\left| \overrightarrow { b } \right| a\quad \bot \quad \left| \overrightarrow { a } \right| \cdot \overrightarrow { b } -\left| \overrightarrow { b } \right| a$ 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 $\overrightarrow { a } \cdot \overrightarrow { a } =0\quad and\quad \overrightarrow { a } \cdot \overrightarrow { b } =0$, then what can be concluded about the vector $\overrightarrow { b }$ ?
Solution:
$\overrightarrow { a } \cdot \overrightarrow { a } =0\quad and\quad \overrightarrow { a } \cdot \overrightarrow { b } =0$
=> $\overrightarrow { b }$ = 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 $\overrightarrow { a } =0\quad or\quad \overrightarrow { b } =0$ then $\overrightarrow { a } .\overrightarrow { b } =0$. But the converse need not be true. Justify your answer with an example.
Solution:
Given: $\overrightarrow { a } =0\quad or\quad \overrightarrow { b } =0$
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.
12

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 $2\hat { i } -\hat { j } +\hat { k } ,\hat { i } -3\hat { j } -5\hat { k }$ and $\left( 3\hat { i } -4\hat { j } -4\hat { k } \right)$ from the vertices of a right angled triangle.
Solution:
The position vectors of the points A, B and C are
$2\hat { i } -\hat { j } +\hat { k } ,\hat { i } -3\hat { j } -5\hat { k }$ and $\left( 3\hat { i } -4\hat { j } -4\hat { k } \right)$

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 : λa→ is a unit vectors

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Exercise 10.3 PDF
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