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## NCERT Solutions class 12 Maths Vector Algebra

Vector Algebra

Exercise 10.4

1. Find  if  and

Ans. Given:  and

Expanding along first row,

=

=

#### 2. Find a unit vector perpendicular to each of the vectors  and  where  and

Ans. Given:  and

On Subtracting   =    =

Therefore,

Expanding along first row =

=

Therefore, a unit vector perpendicular to both   and  is

=  =

#### 3. If a unit vector  makes an angle  with  with  and an acute angle  with  then find  and hence, the components of .

Ans. Let  be a unit vector.  ……….(i)

Squaring both sides,   ……….(ii)

Given: Angle between vectors  and  is

……….(iii)

Again, given Angel between vectors  and  is

……….(iv)

Again, given Angel between vectors  and  is  where  is acute angle.

……….(v)

Putting the values of  and  in eq. (ii),

Since  is acute angle, therefore  is positive and hence

From eq. (v),

Putting values of  and  in eq. (i),

Components of  are coefficients of  in

and angle

#### 4. Show that

Ans. L.H.S. =  =

=  =  = R.H.S.

#### 5. Find  and  if

Ans. Given:

Expanding along first row,

=

Comparing the coefficients of  on both sides, we have

……….(i)

……….(ii)

And   ……….(iii)

From eq. (ii),

From eq. (iii),

Putting the values of  and  in eq. (i),

0 = 0

Therefore,  and

#### 6. Given that  and  What can you conclude about the vectors  and

Ans. Given:

or   or

or   or vector  is perpendicular to  …..(i)

Again, given

or   or

or   or vector  and  are collinear or parallel. …..(ii)

Since, vectors  &  are perpendicular to each other as well as parallel are not possible. ..(iii)

Therefore, from eq. (i), (ii) and (iii),  either   or

and

#### 7. Let the vectors  be given as  then show that

Ans. Given: Vector  and

Now L.H.S. =

=  +

[By Property of Determinants]

=  = R.H.S.

#### 8. It either  and  then  Is the converse true? Justify your answer with an example.

Ans. Given: Either  or

or ……….(i)

[Using eq. (i)]

[By definition of zero vector]

But the converse is not true.

Let

is a non-zero vector.

Let

is a non-zero vector.

But

Taking 2 common from R3 =   [ R2 and R3 are identical]

#### 9. Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).

Ans. Vertices of  are A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).

Position vector of point A = (1, 1, 2) =

Position vector of point B = (2, 3, 5) =

Position vector of point C = (1, 5, 5) =

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

=

=

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

=

=

x  =

=  =

Now Area of triangle ABC =

=  sq. units

#### 10. Find the area of the parallelogram whose adjacent sides are determined by the vectors  and

Ans. Given: Vectors representing two adjacent sides of a parallelogram are

and

=  =

Now Area of parallelogram =

=  sq. units

#### 11. Let the vectors  and  such that  then  is a unit vector, if the angle between  and  is:

(A)

(B)

(C)

(D)

Ans. Given:  and  is a unit vector.

, where  is the angle between  and

Therefore, option (B) is correct.

#### 12. Area of a rectangle having vertices A, B, C and D with position vectors  and  respectively is:

(A)

(B) 1

(C) 2

(D) 4

Ans. Given: ABCD is a rectangle.

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

=

=

AB =

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

=

=

Area of rectangle ABCD = Length x Breadth = AB x AD = 2 x 1 = 2 sq. units

Therefore, option (C) is correct.

## NCERT Solutions class 12 Maths Exercise 10.4

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