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## NCERT Solutions class 12 Maths Vector Algebra

1. Write down a unit vector in XY-plane making an angle of  with the positive direction of axis.

Ans. Let  be the unit vector in XY-plane such that XOP =

Therefore,     OP = 1   ……….(i)

By Triangle Law of Addition of vectors,

In ,  =

[Unit vector along OX is  and that is along OY is ]

[Dividing and multiplying by OP in R.H.S.]

[Using eq. (i)]

#### 2. Find the scalar components and magnitude of the vector joining the points P and Q

Ans. Given points are P and Q

Position vector of point P =  =

And Position vector of point Q =  =

Now   = Position vector of Q – Position vector of P

=

=  =

Scalar components of the vector  are the coefficients of  in , i.e.,

And magnitude of vector  =

#### 3. A girl walks 4 km towards west, then she walks 3 km in a direction  east of north and stops. Determine the girl’s displacement from her initial point of departure.

Ans. Let the initial point of departure is origin (0, 0) and the girl walks a distance OA = 4 km towards west.

Through the point A, draw a line AQ parallel to a line OP, which is  East of North, i.e., in East-North quadrant making an angle of  with North.

Again, let the girl walks a distance AB = 3 km along this direction

=    ……….(i)  [ Vector  is along OX’]

Now, draw BM perpendicular to axis.

In  by Triangle Law of Addition of vectors,

Dividing and multiplying by AB in R.H.S.,

=

……….(ii)

Girl’s displacement from her initial point O of departure to final point B,

=  =

Ans. Given:

Either the vectors  are collinear or form the sides of a triangle.

Case I: Vectors  are collinear.

Let  and

Then

Also,  = AC = AB + BC =

Case II: Vectors  form a triangle.

Here also by Triangle Law of vectors,

But   [ Each side of a triangle is less than sum of the other two sides]

is true only when vectors  and  are collinear vectors.

#### 5. Find the value of  for which  is a unit vector.

Ans. Since  is a unit vector,

Therefore,

Squaring both sides,

#### 6. Find a vector of magnitude 5 units and parallel to the resultant of the vectors  and

Ans. Given: Vectors  and

Let vector  be the resultant vector of  and

=  +

=

Required vector pf magnitude 5 units and parallel (or collinear) to resultant vector  is

=

=

=

=

#### 7. If  and  find a unit vector parallel to the vector

Ans. Given: Vectors  and

Let

=

=

=

A unit vector parallel to the vector  is

=

=

#### 8. Show that the points A B and C (11, 3, 7) are collinear and find the ratio in which B divides AC.

Ans. Given: Points A B and C (11, 3, 7).

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

=  =  =

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

=  =  =

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

=  =  =

Now  =  =

Therefore, points A, B, C are either collinear or are the vertices of a triangle ABC.

Again AB + BC =  = AC

Now to find ratio in which B divides AC

Let the point B divides AC in the ratio

Therefore, using section formula, Position vector of point B  is

=

Comparing coefficients of  both sides, we get

Therefore, required ratio =  =  : 1 = 2 : 3

#### 9. Find the position vector of a point R which divides the line joining the two points P and Q whose position vectors are  and  externally in the ratio 1 : 2. Also, show that P is the middle point of line segment RQ.

Ans. Since position vector of point R dividing the join of P and Q externally in the ratio 1 : 2 =  is given by

Again position vector of the middle point of the line segment RQ

=  (Position vector of point R – Position vector of point Q)

=  =  =  = Position vector of point P (given)

Therefore, P is the middle point of the line segment RQ.

#### 10. Two adjacent sides of a parallelogram are  and  Find the unit vector parallel to its diagonal. Also, find its area.

Ans. Let ABCD is a parallelogram.

Given: The vectors representing two adjacent sides of this parallelogram say,

and

Now vectors along the diagonals  and  of the parallelogram are

and

=  =

And  =  =

Therefore, Unit vectors parallel to (or along) diagonals are

and

and      and

and

Now Area of parallelogram =  =

=  =

=  =  =  sq. units

#### 11. Show that the direction cosines of a vector equally inclined  to the axes OX, OY and OZ are ,,.

Ans. Let  be the direction cosines of a vector equally inclined to axes OX, OY and OZ respectively.

A unit vector along the given vector is

and

……….(i)

Let the given vector (for which unit vector is ) make equal angle (given)  (say) with OX OY and OZ

The given vector is in positive octant OXYZ and hence  is acute. ……….(ii)

Now angle  between  and

……….(iii)

Similarly, angle  between  and ,       ……….(iv)

And angle  between  and ,       ……….(v)

Putting the values of  in eq. (i), we get

But  [  is acute and hence  is positive]

Therefore, required vectors  are  and

#### 12. Let  and  Find a vector  which is perpendicular to both  and  and

Ans. Given: Vectors  and

We know that the cross-product of two vectors,  is a vector perpendicular to both  and

Hence, vector  which is also perpendicular to both  and  is  where  or some other scalar.

Therefore,

=

………..(i)

Now given  and

Putting  in eq. (i), we get

#### 13. The scalar product of the vector  with a unit vector along the sum of vectors  and  is equal to one. Find the value of

Ans. Let ,  and

Now  (say) =

a unit vector along  is

=

=

=

…..(i)

Also given Dot product of  and  is 1.

. = 1

Squaring both sides,

#### 14. If  are mutually perpendicular vectors of equal magnitudes, show that the vector  is equally inclined to .

Ans. Given:  are mutually perpendicular vectors of equal magnitude.

……….(i)

And  (say)        ……….(ii)

Let vector  make angles  with vectors  respectively.

=

=   [From eq. (i)]

=  =       ……….(iii)

We know that

=

Putting the values from eq. (i) and (ii),

=  =

Now  =

=

Similarly,  and

Therefore,  is equally inclined to the vectors  and

#### 15. Prove that  if and only if  are perpendicular given

Ans. We know that  =

=      ……….(i)

Now if  and  are perpendicular

Putting  in

= ,

=       ……….(ii)

=  [Putting value of  in eq. (i)]

But  (given)

Therefore, vectors  and  are perpendicular to each other.

#### 16. Choose the correct answer:

If  is the angle between two vectors  and  then  only when:

(A)

(B)

(C)

(D)

Ans. Given:

[ and  being lengths of vectors are always  0]

Therefore, option (B) is correct.

#### 17. Choose the correct answer:

Let  and  be two unit vectors and  is the angle between them. Then  is a unit vector if:

(A)

(B)

(C)

(D)

Ans. Given:  and  are unit vectors.

and

Now squaring both sides of , we have,

, where  is the given angle between vectors  and .

Putting , we have,

=

=  =

Therefore, option (D) is correct.

#### 18. Choose the correct answer:

The value of  is:

(A) 0

(B)

(C) 1

(D) 3

Ans.

Also  = 1 – 1 + 1

Therefore, option (C) is correct.

#### 19. If  be the angle between any two vectors  and , then  when  is equal to:

(A) 0

(B)

(C)

(D)

Ans. Given:

And this equation is true only for option (B) namely , since

Therefore, option (B) is correct.

## NCERT Solutions class 12 Maths Miscellaneous

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