Solutions of chapter 10. Vector Algebra | Mathematics Part-II

1

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

2

Classify the following measures as scalars and vectors.

(i) 10 kg (ii) 2 meters north-west

(iii) 40° (iv) 40 watt

(v) 10^-19 coulomb (vi) 20 m/s²

3

Classify the following as scalar and vector quantities.

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

(iv) velocity (v) work done

4

In Fig 10.6 (a square), identify the following vectors.

(i) Coinitial

(ii) Equal

(iii) Collinear but not equal

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.

1

Compute the magnitude of the following vectors:

2

Write two different vectors having same magnitude.

3

Write two different vectors having same direction.

4

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

5

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

6

Find the sum of the vectors

7

Find the unit vector in the direction of the vector

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.

9

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

10

Find a vector in the direction of vector which has magnitude 8 units.

11

Show that the vectors are collinear.

12

Find the direction cosines of the vector

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.

14

Show that the vector is equally inclined to the axes OX, OY and OZ.

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 ratio 2 : 1

(i) internally (ii) externally

16

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

17

Show that the points A, B and C with position vectors, respectively form the vertices of a right angled triangle.

18

In triangle ABC (Fig 10.18), which of the following is not true:

(a)

(b)

(c)

(d)

19

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

(a)

(b)

(c) the respective components of are not proportional

(d) both the vectors have same direction, but different magnitudes.

1

Find the angle between two vectors with magnitudes √3 and 2 respectively having

2

Find the angle between the vectors

3

Find the projection of the vector on the vector

4

Find the projection of the vector on the vector

5

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

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

6

Find if

7

Evaluate the product

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 1/2.

9

Find if for a unit vector

10

If are such that is perpendicular to then find the value of λ.

11

Show that is perpendicular to for any two nonzero vectors .

12

If and then what can be concluded about the vector

13

If are unit vectors such that find the value of

14

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

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 ].

16

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

17

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

18

If is a nonzero vector of magnitude ‘a’ and λ a nonzero scalar, then is unit vector if

1

Find if

2

Find a unit vector perpendicular to each of the vector

3

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

4

Show that

5

Find λ and μ if

6

Given that What can you conclude about the vectors ?

7

Let the vectors be given as Then show that

8

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

9

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

10

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

11

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

12

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

1

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

2

Find the scalar components and magnitude of the vector joining the points P(x₁, y₁, z₁) and Q(x₂, y₂, z₂).

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.

4

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

5

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

6

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

7

If , and , find a unit vector parallel to the vector .

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.

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 midpoint of the line segment RQ.