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

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3.

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

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 } ,
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q1.1

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 } ,
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q2.1

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 }
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q3.1

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
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q4.1

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)
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q5.1

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
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q6.1

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 \frac { 1 }{ 2 }
Solution:
We know that \overrightarrow { a } .\overrightarrow { b } =\left| \overrightarrow { a } \right| \left| \overrightarrow { b } \right| cos\theta
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q8.1

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
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q9.1

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 }
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q10.1

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
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q11.1

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 } \overrightarrow { a } =0\quad and\quad \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
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q13.1

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
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q14.1

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 }
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q15.1

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
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q16.1

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)
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q17.1

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
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.3 Q18.1

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