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 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
\(\overrightarrow { a } =3\hat { i } +2\hat { j } +2\hat { k } \quad and\quad \overrightarrow { b } =\hat { i } +2\hat { j } -2\hat { k } \)
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:
\(Let\quad \overrightarrow { a } ={ a }_{ 1 }\hat { i } +{ a }_{ 2 }\hat { j } +{ a }_{ 3 }\hat { k } such\quad that\quad \left| \overrightarrow { a } \right| =1\)
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 = \(\left( \overrightarrow { a } -\overrightarrow { b } \right) \times \left( \overrightarrow { a } +\overrightarrow { b } \right) \)
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:
\(\left( 2\hat { i } +6\hat { j } +27\hat { k } \right) \times \left( \hat { i } +\lambda \hat { j } +\mu \hat { k } \right) =0\)
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:
\(\overrightarrow { a } .\overrightarrow { b } =0\quad and\quad \overrightarrow { a } \times \overrightarrow { b } =0\)
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 \({ a }_{ 1 }\hat { i } +{ a }_{ 2 }\hat { j } +{ a }_{ 3 }\hat { k } ,{ b }_{ 1 }\hat { i } +{ b }_{ 2 }\hat { j } +{ b }_{ 3 }\hat { k } ,{ c }_{ 1 }\hat { i } +{ c }_{ 2 }\hat { j } +{ c }_{ 3 }\hat { k } \)
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:
\(\overrightarrow { a } =0\Rightarrow \left| \overrightarrow { a } \right| =0\)
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).
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 \(\overrightarrow { a } =\hat { i } -\hat { j } +3\hat { k } ,\overrightarrow { b } =2\hat { i } -7\hat { j } +\hat { k } \)
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
\(\left| \overrightarrow { a } \times \overrightarrow { b } \right| =1\)
\(\left| \overrightarrow { a } \right| =3,\left| \overrightarrow { b } \right| =\frac { \sqrt { 2 } }{ 3 } \)
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:
\(\overrightarrow { OA } =\left( -\hat { i } +\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right) \)
\(\overrightarrow { OB } =\left( \hat { i } +\frac { 1 }{ 2 } \hat { j } +4\hat { k } \right) \)
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