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# NCERT Solutions class 12 Maths Exercise 10.3

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## 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),

Ans. Given:  =

=

=

=

#### 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

#### 9. Find  if for a unit vector

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

Putting  from eq. (i),

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

Ans. Given: ,  and

Now  =

Again, =

Since,  is perpendicular to  therefore, .

#### 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.

#### 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.

#### 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

#### 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 .

#### 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)]

#### 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.

#### 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)

+  =  +  =  =  [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.

#### 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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