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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 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

Ex 10.4 Class 12 Maths 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

NCERT Maths Class 12 Chapter 10

Ex 10.4 Class 12 Maths 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
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 2.1

Ex 10.4 Class 12 Maths 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

Ex 10.4 Class 12 Maths 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)
vedantu class 12 maths Chapter 10 Vector Algebra 4

Ex 10.4 Class 12 Maths 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

Ex 10.4 Class 12 Maths 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

Ex 10.4 Class 12 Maths 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 }
vedantu class 12 maths Chapter 10 Vector Algebra 7
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 7.1

Ex 10.4 Class 12 Maths 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

Ex 10.4 Class 12 Maths 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
NCERT Solutions for Class 12 Maths Chapter 10 Vector Algebra 9.1

Ex 10.4 Class 12 Maths 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 }
vedantu class 12 maths Chapter 10 Vector Algebra 10

Ex 10.4 Class 12 Maths 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

Ex 10.4 Class 12 Maths 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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