Exercise 10.1

Question 1:

Represent graphically a displacement of 40 km, 30° east of north. Answer

Here, vector  represents the displacement of 40 km, 30° East of North.

Question 2:

Classify the following measures as scalars and vectors.

(i) 10 kg (ii) 2 metres north-west (iii) 40°

(iv) 40 watt (v) 10^–19 coulomb (vi) 20 m/s² Answer

(i)  10 kg is a scalar quantity because it involves only magnitude.

(ii)  2 meters north-west is a vector quantity as it involves both magnitude and direction.

(iii)  40° is a scalar quantity as it involves only magnitude.

(iv)  40 watts is a scalar quantity as it involves only magnitude.

(v)  10^–19 coulomb is a scalar quantity as it involves only magnitude.

(vi)  20 m/s² is a vector quantity as it involves magnitude as well as direction.

Question 3:

Classify the following as scalar and vector quantities.

(i) time period (ii) distance (iii) force

(iv) velocity (v) work done Answer

(i)  Time period is a scalar quantity as it involves only magnitude.

(ii)  Distance is a scalar quantity as it involves only magnitude.

(iii)  Force is a vector quantity as it involves both magnitude and direction.

(iv)  Velocity is a vector quantity as it involves both magnitude as well as direction.

(v)  Work done is a scalar quantity as it involves only magnitude.

Question 4:

(i) Coinitial (ii) Equal (iii) Collinear but not equal Answer

(i)  Vectors    and  are coinitial because they have the same initial point.

(ii)  Vectors   and  are equal because they have the same magnitude and direction.

(iii)  Vectors    and  are collinear but not equal. This is because although they are parallel, their directions are not the same.

Question 5:

Answer the following as true or false.

(i)  and    are collinear.

(ii)  Two collinear vectors are always equal in magnitude.

(iii)  Two vectors having same magnitude are collinear.

(iv)  Two collinear vectors having the same magnitude are equal.

Answer

(i)  True.

Vectors  and  are parallel to the same line.

(ii)  False.

Collinear vectors are those vectors that are parallel to the same line.

(iii)  False.

Exercise 10.2

Question 1:

Compute the magnitude of the following vectors:

Answer

The given vectors are:

Question 2:

Hence,   are two different vectors having the same magnitude. The vectors are different because they have different directions.

Question 3:

Write two different vectors having same direction. Answer

The direction cosines of  are the same. Hence, the two vectors have the same direction.

Question 4:

Find the values of x and y so that the vectors  are equal Answer

The two vectors    will be equal if their corresponding components are equal.

Hence, the required values of x and y are 2 and 3 respectively.

Question 5:

Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (–5, 7).

Answer

The vector with the initial point P (2, 1) and terminal point Q (–5, 7) can be given by,

Hence, the required scalar components are –7 and 6 while the vector components are

Question 6:

Find the sum of the vectors .

Answer

The given vectors are .

Question 7:

Find the unit vector in the direction of the vector . Answer

The unit vector    in the direction of vector                      is given by .

Question 8:

Find the unit vector in the direction of vector , where P and Q are the points (1, 2, 3) and (4, 5, 6), respectively.

Answer

The given points are P (1, 2, 3) and Q (4, 5, 6).

Hence, the unit vector in the direction of  is

.

Question 9:

For given vectors,  and  , find the unit vector in the direction of the vector

Answer

The given vectors are  and .

Hence, the unit vector in the direction of   is

.

Question 10:

Question 11:

Show that the vectors  are collinear. Answer

.

Hence, the given vectors are collinear.

Question 12:

 

Hence, the direction cosines of

Question 13:

Find the direction cosines of the vector joining the points A (1, 2, –3) and B (–1, –2, 1) directed from A to B.

Answer

The given points are A (1, 2, –3) and B (–1, –2, 1).

Hence, the direction cosines of  are

Question 14:

Therefore, the direction cosines of

Now, let α, β, and γbe the angles formed by  with the positive directions of x, y, and z

axes.

Then, we have

Hence, the given vector is equally inclined to axes OX, OY, and OZ.

Question 15:

Find the position vector of a point R which divides the line joining two points P and Q

whose position vectors are  respectively, in the ration 2:1

(i)  internally

(ii)  externally Answer

The position vector of point R dividing the line segment joining two points P and Q in the ratio m: n is given by:

i.        Internally:

ii.        Externally:

Position vectors of P and Q are given as:

(i)  The position vector of point R which divides the line joining two points P and Q internally in the ratio 2:1 is given by,

(ii) 

Question 16:

Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, – 2).

Answer

Question 17:

Show that the points A, B and C with position vectors,                         ,

 respectively form the vertices of a right angled triangle. Answer

Position vectors of points A, B, and C are respectively given as:

Hence, ABC is a right-angled triangle.

Question 18:

A.

B.

C.

D.

On applying the triangle law of addition in the given triangle, we have:

From equations (1) and (3), we have:

Hence, the equation given in alternative C is incorrect. The correct answer is C.

Question 19:

If  are two collinear vectors, then which of the following are incorrect:

A.    , for some scalar λ

B.

C. the respective components of  are proportional

D. both the vectors    have same direction, but different magnitudes Answer

If  are two collinear vectors, then they are parallel. Therefore, we have:

 (For some scalar λ)

If λ = ±1, then  .

Thus, the respective components of   are proportional. However, vectors  can have different directions.

Hence, the statement given in D is incorrect.

The correct answer is D.

Exercise 10.3

Question 1:

Find the angle between two vectors  and   with magnitudes    and 2, respectively having  .

Answer

It is given that,

Hence, the angle between the given vectors     and    is  .

Question 2:

Find the angle between the vectors  Answer

The given vectors are                                                    .

Also, we know that

.

Question 3:

Find the projection of the vector  on the vector . Answer

Let  and  .

Now, projection of vector     on   is given by,

Hence, the projection of vector  on   is 0.

Question 4:

Find the projection of the vector  on the vector  . Answer

Let  and  .

Now, projection of vector    on  is given by,

Question 5:

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

Also, show that they are mutually perpendicular to each other. Answer

Thus, each of the given three vectors is a unit vector.

Hence, the given three vectors are mutually perpendicular to each other.

Question 6:

Find      and  , if  .

Answer

Question 7:

Evaluate the product . Answer

Question 8:

Find the magnitude of two vectors , having the same magnitude and such that

the angle between them is 60° and their scalar product is . Answer

Let θ be the angle between the vectors  It is given that

We know that

.

Question 9:

Question 10:

If  are such that  is perpendicular to  ,

Hence, the required value of λ is 8.

Question 11:

Show that  is perpendicular to  , for any two nonzero vectors Answer

Hence,  and     are perpendicular to each other.

Question 12:

If  , then what can be concluded about the vector  ?

Answer

It is given that .

Hence, vector      satisfying  can be any vector.

Question 14:

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

Answer

We now observe that:

Hence, the converse of the given statement need not be true.

Question 15:

If the vertices A, B, C of a triangle ABC are (1, 2, 3), (–1, 0, 0), (0, 1, 2), respectively,

then find ∠ABC. [∠ABC is the angle between the vectors  and      ] Answer

The vertices of ∆ABC are given as A (1, 2, 3), B (–1, 0, 0), and C (0, 1, 2). Also, it is given that ∠ABC is the angle between the vectors    and     .

Now, it is known that:

 .

Question 16:

Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear. Answer

The given points are A (1, 2, 7), B (2, 6, 3), and C (3, 10, –1).

Hence, the given points A, B, and C are collinear.

Question 17:

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

Answer

Now, vectors  represent the sides of ∆ABC.

Hence, ∆ABC is a right-angled triangle.

Question 18:

If  is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then λ   is unit vector if (A) λ = 1 (B) λ = –1 (C)                                              (D)   

Answer

Vector      is a unit vector if            .

Hence, vector       is a unit vector if           . The correct answer is D.

Exercise 10.4

Question 1:

Find  , if                        and                         . Answer

We have,

and

Question 2:

Find a unit vector perpendicular to each of the vector  and , where  and    .

Answer We have,

and

and

Question 3:

If a unit vector    makes an angles     with with     and an acute angle θ with , then find θ and hence, the compounds of .

Answer

Let unit vector  have (a₁, a₂, a₃) components.

Since  is a unit vector,  .

Also, it is given that     makes angles      with with     , and an acute angle θ with Then, we have:

Hence,          and the components of   are                      .

Question 4:

Show that

Answer

Question 5:

Find λ and µ if                                                     .

Answer

On comparing the corresponding components, we have:

Hence,

Question 6:

Then,

(i) 

Either

or , or

(ii)  Either    or , or  But,  and     cannot be perpendicular and parallel simultaneously.

Hence,           or          .

Question 7:

Let the vectors  given as   . Then show that

Answer We have,

On adding (2) and (3), we get:

Now, from (1) and (4), we have:

Hence, the given result is proved.

Question 8:

If either  or , then  . Is the converse true? Justify your answer with an example.

Answer

Take any parallel non-zero vectors so that .

It can now be observed that:

Hence, the converse of the given statement need not be true.

Question 9:

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

C (1, 5, 5).

Answer

The vertices of triangle ABC are given as A (1, 1, 2), B (2, 3, 5), and

C (1, 5, 5).

The adjacent sides     and         of ∆ABC are given as:

Hence, the area of ∆ABC

Question 10:

Find the area of the parallelogram whose adjacent sides are determined by the vector

.

Answer

Hence, the area of the given parallelogram is  .

Question 11:

Let the vectors    and     be such that            and , then         is a unit vector, if the angle between  and       is

(A) (B) (C) (D)  Answer

It is given that

.

We know that , where    is a unit vector perpendicular to both     and  and θ is the angle between  and   .

Hence,         is a unit vector if the angle between     and     is . The correct answer is B.

Question 12:

Area of a rectangle having vertices A, B, C, and D with position vectors

 and  respectively is

(A)  (B) 1

(C) 2 (D) Answer

The position vectors of vertices A, B, C, and D of rectangle ABCD are given as:

The adjacent sides     and   of the given rectangle are given as:

Now, it is known that the area of a parallelogram whose adjacent sides are   is .

Hence, the area of the given rectangle is  The correct answer is C.

Miscellaneous Solutions

Question 1:

Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

Answer

If     is a unit vector in the XY-plane, then

Hence, the required unit vector is

Question 2:

Find the scalar components and magnitude of the vector joining the points

.

Answer

The vector joining the points                                                can be obtained by,

Hence, the scalar components and the magnitude of the vector joining the given points are respectively   and  .

Question 3:

A girl walks 4 km towards west, then she walks 3 km in a direction 30° east of north and stops. Determine the girl’s displacement from her initial point of departure.

Answer

Let O and B be the initial and final positions of the girl respectively. Then, the girl’s position can be shown as:

Now, we have:

By the triangle law of vector addition, we have:

Hence, the girl’s displacement from her initial point of departure is

.

Question 4:

If               , then is it true that ? Justify your answer. Answer

Now, by the triangle law of vector addition, we have  .

It is clearly known that    represent the sides of ∆ABC.

Hence, it is not true that .

Question 5:

Find the value of x for which  is a unit vector. Answer

is a unit vector if

.

Hence, the required value of x is         .

Question 6:

Find a vector of magnitude 5 units, and parallel to the resultant of the vectors

.

Let  be the resultant of .

Hence, the vector of magnitude 5 units and parallel to the resultant of vectors    is

Question 7:

If  , find a unit vector parallel to the

vector .

Answer We have,

Hence, the unit vector along     is

Question 8:

Show that the points A (1, –2, –8), B (5, 0, –2) and C (11, 3, 7) are collinear, and find the ratio in which B divides AC.

Answer

The given points are A (1, –2, –8), B (5, 0, –2), and C (11, 3, 7).

Thus, the given points A, B, and C are collinear.

Now, let point B divide AC in the ratio . Then, we have:

On equating the corresponding components, we get:

Hence, point B divides AC in the ratio

Question 9:

Find the position vector of a point R which divides the line joining two points P and Q

whose position vectors are     externally in the ratio 1: 2. Also, show that P is the mid point of the line segment RQ.

Answer

It is given that .

It is given that point R divides a line segment joining two points P and Q externally in the ratio 1: 2. Then, on using the section formula, we get:

Therefore, the position vector of point R is   . Position vector of the mid-point of RQ =

Hence, P is the mid-point of the line segment RQ.

Question 10:

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

Answer

Adjacent sides of a parallelogram are given as:    and  Then, the diagonal of a parallelogram is given by .

Thus, the unit vector parallel to the diagonal is

Hence, the area of the parallelogram is  square units.

Question 11:

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

are  Answer

Let a vector be equally inclined to axes OX, OY, and OZ at angle α. Then, the direction cosines of the vector are cos α, cos α, and cos α.

Hence, the direction cosines of the vector which are equally inclined to the axes

are

.

Question 12:

Let    and  . Find a vector  which is perpendicular to both   and  , and .

Answer

Let .

Since  is perpendicular to both  and  , we have:

Also, it is given that:

On solving (i), (ii), and (iii), we get:

Hence, the required vector is .

Question 13:

The scalar product of the vector              with a unit vector along the sum of vectors

Scalar product of                with this unit vector is 1.

Hence, the value of λ is 1.

Question 14:

If  are mutually perpendicular vectors of equal magnitudes, show that the vector

 is equally inclined to  and  . Answer

Since     are mutually perpendicular vectors, we have

It is given that:

Hence, the vector  is equally inclined to                  .

Question 15:

Prove that , if and only if          are perpendicular, given .

Answer

Question 16:

If θ is the angle between two vectors   and   , then   only when

(A)  (B)

(C)  (D)

Answer

Let θ be the angle between two vectors  and  .

Then, without loss of generality,    and   are non-zero vectors so that         .

Hence,            when  . The correct answer is B.

Question 17:

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

(A)  (B)  (C)  (D)  Answer

Let   and   be two unit vectors andθ be the angle between them.

Then,

.

Now,  is a unit vector if               .

Hence,          is a unit vector if . The correct answer is D.

Question 18:

The value of  is

(A) 0 (B) –1 (C) 1 (D) 3

Answer

The correct answer is C.

Question 19:

If θ is the angle between any two vectors    and  , then                    when θ isequal to

(A) 0 (B)    (C)   (D) π Answer

Let θ be the angle between two vectors  and  .

Then, without loss of generality,   and  are non-zero vectors, so that

.

Hence,   when θ isequal to The correct answer is B.