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NCERT Solutions class 12 Maths Exercise 10.3 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

**Download NCERT solutions for Vector Algebra as PDF.**

## NCERT Solutions class 12 Maths Vector Algebra

**1. Find the angle between two vectors **** and **** with magnitude **** and 2 respectively having **

**Ans. **Given: and

Let be the angle between the vector and

We know that

= =

**2. Find the angle between the vectors **** and **** **

**Ans. **Given: Let and

and

Also

= Product of coefficients of + Product of coefficients of + Product of coefficients

=

Let be the angle between the vector and

We know that

= =

**3. Find the projection of the vector **** on the vector **** **

**Ans. **Let and

Projection of vector and =

=

=

If projection of vector and is zero, then vector is perpendicular to

**4. Find the projection of the vector **** on the vector **** **

**Ans. **Let and

Projection of vector and =

=

=

**5. Show that each of the given three vectors is a unit vector:**

** **

**Also show that they are mutually perpendicular to each other.**

**Ans. **Let ……….(i)

……….(ii)

……….(iii)

Each of the three given vectors is a unit vector.

From eq. (i) and (ii),

=

and are perpendicular to each other.

From eq. (ii) and eq. (iii),

=

and are perpendicular to each other.

From eq. (i) and (iii),

=

and are perpendicular to each other.

Hence, are mutually perpendicular vectors.

**6. Find **** and **** if **** and **** **

**Ans. **Given: and ……….(i)

……….(ii)

Putting in eq. (ii),

Putting in eq (i),

**7. Evaluate the product **** **

**Ans. **Given: =

=

=

=

#### NCERT Solutions class 12 Maths Exercise 10.3

**8. Find the magnitude of two vectors **** and **** having the same magnitude such that the angle between them is **** and their scalar product is **** **

**Ans. **Given: , angle (say) between and is and their scalar (i.e., dot) product =

Putting and we have

=

and

#### NCERT Solutions class 12 Maths Exercise 10.3

**9. Find **** if for a unit vector **** **** **

**Ans. **Given: is a unit vector ……….(i)

Putting from eq. (i),

#### NCERT Solutions class 12 Maths Exercise 10.3

**10. If **** and **** are such that **** is perpendicular to **** then find the value of **** **

**Ans. **Given: , and

Now =

Again, =

Since, is perpendicular to therefore, .

#### NCERT Solutions class 12 Maths Exercise 10.3

**11. Show that **** is perpendicular to **** for any two non-zero vectors **** and **** **

**Ans. **Let , where and

Let

Now

=

=

=

Putting, and ,

= = 0

Therefore, vectors and are perpendicular ot each other.

#### NCERT Solutions class 12 Maths Exercise 10.3

**12. If **** and ****, then what can be concluded about the vector **** **

**Ans. **Given:

Again

0 = 0 for all (any vector )

Therefore, can be any vector.

#### NCERT Solutions class 12 Maths Exercise 10.3

**13. If **** are unit vectors such that **** find the value of **

**Ans. **Since, are unit vectors.

Therefore, and ……….(i)

Also given

Putting the values from eq. (i), we get

#### NCERT Solutions class 12 Maths Exercise 10.3

**14. If either vector **** or **** then ****. But the converse need not be true. Justify your answer with an example.**

**Ans. ****Case **I: Vector . Therefore by definition of zero vector, ……….(i)

= [From eq. (i)]

**Case II**: Vector . Therefore by definition of zero vector, ……….(ii)

= [From eq. (ii)]

But the converse is not true.

**Justification**: Let

Therefore,

Therefore,

Again let

Therefore,

But

Hence, here , but and .

#### NCERT Solutions class 12 Maths Exercise 10.3

**15. If the vertices A, B, C of a triangle ABC are **** and (0, 1, 2) respectively, then find **** **

**Ans. **Vertices A, B, C of a triangle are A (1, 2,3 ), B and C (0, 1, 2) respectively.

Position vector of point A =

Position vector of point B =

Position vector of point C =

Now = Position vector of point A – Position vector of point B

=

= ……….(i)

And = Position vector of point C – Position vector of point B

=

= ……….(ii)

Let be the angle between the vectors and .

= [Using eq. (i) and (ii)]

#### NCERT Solutions class 12 Maths Exercise 10.3

**16. Show that the points A (1, 2, 7), B (2, 6, 3) and C**** are collinear.**

**Ans. **Vertices A, B, C of a triangle are A (1, 2, 7), B (2, 6, 3) and C respectively.

Position vector of point A =

Position vector of point B =

Position vector of point C =

Now = Position vector of point B – Position vector of point A

=

= ……….(i)

And = Position vector of point C – Position vector of point A

=

= ……….(ii)

= 2. [Using eq. (i)]

Vectors and are collinear and parallel.

Thus, points A, B and C are collinear.

And also vectors and have a common point A and hence can’t be parallel.

#### NCERT Solutions class 12 Maths Exercise 10.3

**17. Show that the vectors **** and **** form the vertices of a right angled triangle.**

**Ans. **Let the given position vectors be A, B, C.

Position vector of point A is Position vector of point B is and Position vector of point C is

= Position vector of B – Position vector of A

=

= = ……….(i)

= Position vector of C – Position vector of B

=

= = ……….(ii)

= Position vector of C – Position vector of A

=

= = ……….(iii)

Adding eq. (i) and (ii),

+ = + = = [Using eq. (iii)]

Therefore, by Triangle law of addition of vectors, points A, B, C are the vertices of a triangle ABC.

Now from eq. (i) and (ii),

. = =

Again from eq. (ii) and (iii),

. = = 2 + 3 – 5 = 0

is perpendicular to .

Angle C is Therefore is a right angled at C.

Thus, A, B, C are the vertices of a right angled triangle.

#### NCERT Solutions class 12 Maths Exercise 10.3

**18. If **** is a non-zero vector of magnitude **** and **** is a non-zero scalar, then **** is a unit vector if:**

**(A) **

**(B) **

**(C) **

**(D) **

**Ans. **Given: is a non-zero vector of magnitude

Also given and is a unit vector.

Therefore, option (D) is correct.

## NCERT Solutions class 12 Maths Exercise 10.3

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

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