# NCERT Solutions for Class 12 Maths Maths Part-2 Chapter 10

## Vector Algebra Class 12

### Exercise 10.1 : Solutions of Questions on Page Number : 428

Q1 :

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

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

Q2 :

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

(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.

Q3 :

Classify the following as scalar and vector quantities.

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

(iv) velocity (v) work done

Q4 :

In Figure, identify the following vectors.

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

Q5 :

Answer the following as true or false.

(i) andare 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.

### Exercise 10.2 : Solutions of Questions on Page Number : 440

Q1 :

Compute the magnitude of the following vectors:

Q2 :

Write two different vectors having same magnitude.

Q3 :

Write two different vectors having same direction.

Q4 :

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

Q5 :

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

Q6 :

Find the sum of the vectors.

Q7 :

Find the unit vector in the direction of the vector.

Q8 :

Find the unit vector in the direction of vector, where P and Q are the points

(1, 2, 3) and (4, 5, 6), respectively.

Q9 :

Find the unit vector in the direction of vector, where P and Q are the points

(1, 2, 3) and (4, 5, 6), respectively.

Q10 :

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

Q11 :

Show that the vectorsare collinear.

Q12 :

Find the direction cosines of the vector

Q13 :

Find the direction cosines of the vector joining the points A (1, 2, -3) and

B (-1, -2, 1) directed from A to B.

Q14 :

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

Q15 :

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

Q16 :

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

Q17 :

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

Q18 :

In triangle ABC which of the following is not true:

A.

B.

C.

D.

Q19 :

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

A. , for some scalar ÃŽÂ»

B.

C. the respective components of are proportional

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

### Exercise 10.3 : Solutions of Questions on Page Number : 447

Q1 :

Find the angle between two vectorsandwith magnitudesand 2, respectively having.

Q2 :

Find the angle between the vectors

Q3 :

Find the projection of the vectoron the vector.

Q4 :

Find the projection of the vectoron the vector.

Q5 :

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

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

Q6 :

Findand, if.

Q7 :

Evaluate the product.

Q8 :

Find the magnitude of two vectors, having the same magnitude and such that the angle between them is 60° and their scalar product is.

Q9 :

Find, if for a unit vector.

Q10 :

Ifare such thatis perpendicular to, then find the value of ÃŽÂ».

Q11 :

Show that is perpendicular to, for any two nonzero vectors

Q12 :

If, then what can be concluded about the vector?

Q13 :

If are unit vectors such that , find the value of .

Q14 :

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

Q15 :

If the vertices A, B, C of a triangle ABC are (1, 2, 3), (â€“1, 0, 0), (0, 1, 2), respectively, then find ∠ABC. [∠ABC is the angle between the vectorsand]

Q16 :

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

Q17 :

Show that the vectorsform the vertices of a right angled triangle.

Q18 :

Ifis a nonzero vector of magnitude 'a' and ÃŽÂ» a nonzero scalar, then ÃŽÂ»is unit vector if

(A) ÃŽÂ» = 1 (B) ÃŽÂ» = â€“1 (C)

(D)

### Exercise 10.4 : Solutions of Questions on Page Number : 454

Q1 :

Find, if and.

Q2 :

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

Q3 :

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

Q4 :

Show that

Q5 :

Find ÃŽÂ» and ÃŽÂ¼ if .

Q6 :

Given that and. What can you conclude about the vectors?

Q7 :

Let the vectors given as . Then show that

Q8 :

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

Q9 :

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

C (1, 5, 5).

Q10 :

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

Q11 :

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

(A) (B) (C) (D)

Q12 :

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

(A) (B) 1

(C) 2 (D)

### Exercise Miscellaneous : Solutions of Questions on Page Number : 458

Q1 :

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

Q2 :

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

.

Q3 :

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.

Q4 :

Q5 :

Find the value of x for whichis a unit vector.

Q6 :

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

.

Q7 :

If, find a unit vector parallel to the vector.

Q8 :

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.

Q9 :

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

Q10 :

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

Q11 :

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

Q12 :

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

Q13 :

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

Q14 :

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

Q15 :

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

(A) (B)

(C) (D)

Q16 :

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

(A) (B) (C) (D)

Q17 :

The value of is

(A) 0 (B) â€“1 (C) 1 (D) 3

Q18 :

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

(A) 0 (B) (C) (D) π

### Exercise Miscellaneousmiscellaneous : Solutions of Questions on Page Number : 458

Q1 :

The two adjacent sides of a parallelogram areand .

Find the unit vector parallel to its diagonal. Also, find its area.