NCERT Solutions for class 12th Mathematics Chapter 10 Vector Algebra

Exercise 10.1

Question 1. Represent graphically a displacement of 40km, 30° east of north.

Question 2. Classify the following measures as scalars and vectors.

(i) 10 kg (ii) 2 metres north- west (iii) 40° (v) 10–19 coulombs (iv) 40 watt (vi) 20 m/sec2.

Sol. (i) Mass-scalar (ii) Directed distance-vector (iii) Temperature-scalar (iv) Rate of electricity-scalar (v) Electric charge-vector (vi) Acceleration-vector.

Question 3. Classify the following as scalar and vector quantities (i) time period (ii)distance (iii) force (iv) velocity (v) work.

Sol. Scalar Quantity : (i) time period (ii) distance (v) work.Vector Quantity : (iii) force (iv) velocity

Question 4. In a square, identify the following vectors (i) Co-initial (ii) Equal (iii) collinear but not equal

Note: Two vectors are said to be collinear if they are either in same direction or opposite directions.

Question 5. Answer the following as true or false:

(i) a and a 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.

Sol. (i) True (ii) False (iii) False (iv) False

Exercise 10.2

Question 1. Compute the magnitude of the following vectors:

Question 2. Write two different vectors having same magnitude

Question 3. Write two different vectors having same direction

Note: Two vectors have same direction if their direction cosines are same.

Question 4. Find the values of x and y so that the vectors 2 i 3 j +r rand xi +yj are equal.

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

Question 6. Find the sum of three vectors:

Question 7. Find the unit vector in the direction of the vector a= i +j +2k

Question 8. Find the unit vector in the direction of vector PQ. Where P and Q are the points (1, 2, 3) and (4, 5, 6) respectively.

Question 9. For given vectors a= i +j +k and b= i+ j+ k , find the unit vector in the direction of the vector a + b

Question 10. Find a vector in the direction of 5i- j +2k which has magnitude 8 units.

Question 11. Show that the vector 2i- 3j +4k and -4i +6j- 8k are collinear.

Question 12. Find the direction cosines of the vector i+ 2j +3k .

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.

Question 14. Show that the vector i +j +k are equally inclined to the axes OX, OY, OZ.

Question 15. Find the position vector of a point R which divides the line joining the points whose positive vector are P (i+ 2j -k) and Q ( -i +j +k) in the ratio 2 : 1(i) internally (ii) externally

Question 16. Find the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, –2).

Question 17. Show that the points A, B and C with position vector a= 3i-4j -4k, – b =2i -j +k and c= i- 3j-5k , respectively form the vertices of a right angled triangle

Question 18. In triangle ABC (fig.), which of the following is not true:

Question 19. If a and b are two collinear vectors then which of the following are incorrect:

Sol. (d) Options (d) is incorrect since both the vectors a and b, being collinear, are not necessarily in the same direction. They may have opposite directions. Their magnitudes may be different.

Exercise 10.3

Question 1. Find the angle between two vectors a and b with magnitudes root 3 and 2 respectively, and such that a .b = root 6

Question 2. Find the angle between the vectors i-2 j +3k and 3i- 2j + k

Question 3. Find the projection of the vector i- j on the line represented by the vector i + j.

Question 4. Find the projection of the vector i +3j +7k on the vector 7i -j +8k .

Question 5. Show that each of the given three vectors is a unit vector :1/7(2i 3j 6k),1/7(3i 6j 2k),1/7(6i 2j 3k). Also show that they are mutually perpendicular to each other.

Question 6. Find | a | r and | b | if (a+b).(a-b)and | a | = 8| b |.

Question 7. Evaluate the product : (3 a-5b ).(2a+ 7b ).

Question 8. Find the magnitude of two vectors a and b having the same magnitude and such that the angle between them is 60° and their scalar product is 1/2.

Question 9. Find | x |, if for a unit vector.

Question 10. If a =2i +2j +3k , b = – i +2j +k and c = 3i +j such that a +lambda b is

Question 11. Show that |a | b+|b | a is perpendicular to | a|b- |b | a,for any two non-zero vectors a and b.

Question 12. If a .a = 0 and a b = 0 , then what can be concluded about the vector b?

Question 13. If a, b, c are the unit vector such that a +b +c= 0, then find the value of a.b +b .c +c. a .

Question 14. If either vector a = 0 or b = 0, then a.b =0 But the converse need not be true. Justify your answer with an example.

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 angle ABC.

Question 16. Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, –1) are collinear.

Question 17. Show that the vectors 2i- j+ k, i -3i- 5k, and (3i -4j- 4k) from the vertices of a right-angled triangle.

Question 18. If a is a non-zero vector of magnitude ‘a’ and lamda is a non- zero scalar, then lamda a is unit vector if

Exercise 10.4

Question 1. Find | axb| , if a= i-7j-7k and b=3i-2j +2k

Question 2. Find a unit vector perpendicular to each of the vector a + b and a – b .

Question 3.

Question 4. Show that ( a-b)x (a+b)= 2(axb).

Question 5.

Question 6. Given that a b = 0 and axb = 0. What can you conclude about the vectors a and b?

Question 7.

Question 8. If either a =0 or b= 0, then a xb= 0 . Is the converse true? Justify your answer with an example.

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

Question 10. Find the area of the parallelogram whose adjacent sides are determined by

Question 11.

Question 12. Area of a rectangle having vertices

Miscellaneous Exercise

Question 1. Write down a unit vector in XY-plane, making an angle of 30° with the positive direction of x-axis.

Question 2. Find the scalar components and magnitude of the vector joining the points P (x1, y1, z1) and Q (x2, y2, z2).

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.

Question 4. If a = b + c, then is it true that !a!=! b !+!c!? Justify your answer.

Question 5. Find the value of x for which x (i+ j +k) is a unit vector.

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

Question 7.

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.

Question 9. Find the position vector of a point R which divides the line joining two points P and Q whose position vectors are (2a+ b) and (a-3b) externally in the ratio 1 : 2. Also, show that P is the mid point of the line segment RQ.

Question 10. The two adjacent sides of a parallelogram are 2i- 4j+ 5k and i -2j- 3k .Find the unit vector parallel to its diagonal. Also, find its area.