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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

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