NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4 are part of NCERT Solutions for Class 12 Maths. Here we have given NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4.

Board	CBSE
Textbook	NCERT
Class	Class 12
Subject	Maths
Chapter	Chapter 10
Chapter Name	Vector Algebra
Exercise	Ex 10.4
Number of Questions Solved	12
Category	NCERT Solutions

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4

Question 1.
Find $\left| \overrightarrow { a } \times \overrightarrow { b } \right| ,if\quad \overrightarrow { a } =\hat { i } -7\hat { j } +7\hat { k } \quad and\quad \overrightarrow { b } =3\hat { i } -2\hat { j } +2\hat { k }$
Solution:
Given
$\overrightarrow { a } =\hat { i } -7\hat { j } +7\hat { k } \quad and\quad \overrightarrow { b } =3\hat { i } -2\hat { j } +2\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 1

Question 2.
Find a unit vector perpendicular to each of the vector $\overrightarrow { a } +\overrightarrow { b } \quad and\quad \overrightarrow { a } -\overrightarrow { b }$ , where $\overrightarrow { a } =3\hat { i } +2\hat { j } +2\hat { k } \quad and\quad \overrightarrow { b } =\hat { i } +2\hat { j } -2\hat { k }$
Solution:
we have
$\overrightarrow { a } =3\hat { i } +2\hat { j } +2\hat { k } \quad and\quad \overrightarrow { b } =\hat { i } +2\hat { j } -2\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 2

Question 3.
If a unit vector $\overrightarrow { a }$ makes angle $\frac { \pi }{ 3 } with\quad \hat { i } ,\frac { \pi }{ 4 } with\quad \hat { j }$ and an acute angle θ with $\overrightarrow { k }$ ,then find θ and hence the components of $\overrightarrow { a }$ .
Solution:
$Let\quad \overrightarrow { a } ={ a }_{ 1 }\hat { i } +{ a }_{ 2 }\hat { j } +{ a }_{ 3 }\hat { k } such\quad that\quad \left| \overrightarrow { a } \right| =1$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 3

Question 4.
Show that $\left( \overrightarrow { a } -\overrightarrow { b } \right) \times \left( \overrightarrow { a } +\overrightarrow { b } \right) =2\left( \overrightarrow { a } \times \overrightarrow { b } \right)$
Solution:
LHS = $\left( \overrightarrow { a } -\overrightarrow { b } \right) \times \left( \overrightarrow { a } +\overrightarrow { b } \right)$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 4

Question 5.
Find λ and μ if
$\left( 2\hat { i } +6\hat { j } +27\hat { k } \right) \times \left( \hat { i } +\lambda \hat { j } +\mu \hat { k } \right) =0$
Solution:
$\left( 2\hat { i } +6\hat { j } +27\hat { k } \right) \times \left( \hat { i } +\lambda \hat { j } +\mu \hat { k } \right) =0$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 5

Question 6.
Given that $\overrightarrow { a } .\overrightarrow { b } =0\quad and\quad \overrightarrow { a } \times \overrightarrow { b } =0$ . What can you conclude about the vectors $\overrightarrow { a } ,\overrightarrow { b }$ ?
Solution:
$\overrightarrow { a } .\overrightarrow { b } =0\quad and\quad \overrightarrow { a } \times \overrightarrow { b } =0$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 6

Question 7.
Let the vectors $\overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c }$ are given ${ a }_{ 1 }\hat { i } +{ a }_{ 2 }\hat { j } +{ a }_{ 3 }\hat { k } ,{ b }_{ 1 }\hat { i } +{ b }_{ 2 }\hat { j } +{ b }_{ 3 }\hat { k } ,{ c }_{ 1 }\hat { i } +{ c }_{ 2 }\hat { j } +{ c }_{ 3 }\hat { k }$ . Then show that $\overrightarrow { a } \times \left( \overrightarrow { b } +\overrightarrow { c } \right)$ $=\overrightarrow { a } \times \overrightarrow { b } +\overrightarrow { a } \times \overrightarrow { c }$
Solution:
Given
$\overrightarrow { a } ,\overrightarrow { b } ,\overrightarrow { c }$ are given ${ a }_{ 1 }\hat { i } +{ a }_{ 2 }\hat { j } +{ a }_{ 3 }\hat { k } ,{ b }_{ 1 }\hat { i } +{ b }_{ 2 }\hat { j } +{ b }_{ 3 }\hat { k } ,{ c }_{ 1 }\hat { i } +{ c }_{ 2 }\hat { j } +{ c }_{ 3 }\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 7

Question 8.
If either $\overrightarrow { a } =0\quad or\quad \overrightarrow { b } =0\quad then\quad \hat { a } \times \hat { b } =0$ .Is the
converse true? Justify your answer with an example.
Solution:
$\overrightarrow { a } =0\Rightarrow \left| \overrightarrow { a } \right| =0$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 8

Question 9.
Find the area of the triangle with vertices A (1,1,2), B (2,3,5) and C (1,5,5).
Solution:
A (1,1,2), B (2,3,5) and C (1,5,5).
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 9

Question 10.
Find the area of the parallelogram whose adjacent sides are determined by the vectors $\overrightarrow { a } =\hat { i } -\hat { j } +3\hat { k } ,\overrightarrow { b } =2\hat { i } -7\hat { j } +\hat { k }$
Solution:
We have $\overrightarrow { a } =\hat { i } -\hat { j } +3\hat { k } ,\overrightarrow { b } =2\hat { i } -7\hat { j } +\hat { k }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 10

Question 11.
Let the vectors $\overrightarrow { a } ,\overrightarrow { b }$ such that $\left| \overrightarrow { a } \right| =3,\left| \overrightarrow { b } \right| =\frac { \sqrt { 2 } }{ 3 }$ then $\overrightarrow { a } \times \overrightarrow { b }$ is a unit vector if the angle between $\overrightarrow { a } ,\overrightarrow { b }$ is
(a) $\frac { \pi }{ 6 }$
(b) $\frac { \pi }{ 4 }$
(c) $\frac { \pi }{ 3 }$
(d) $\frac { \pi }{ 2 }$
Solution:
Given
$\left| \overrightarrow { a } \times \overrightarrow { b } \right| =1$
$\left| \overrightarrow { a } \right| =3,\left| \overrightarrow { b } \right| =\frac { \sqrt { 2 } }{ 3 }$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 11

Question 12.
Area of a rectangles having vertices
$A\left( -\hat { i } +\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right) ,B\left( \hat { i } +\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right) ,$
$C\left( \hat { i } -\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right) ,D\left( -\hat { i } -\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right) ,$
(a) $\frac { 1 }{ 2 }$ sq units
(b) 1sq.units
(c) 2sq.units
(d) 4sq.units
Solution:
$\overrightarrow { OA } =\left( -\hat { i } +\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right)$
$\overrightarrow { OB } =\left( \hat { i } +\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right)$
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 12

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NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4

NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra Ex 10.4

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