# 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.

Question 11. Show that the direction cosines of a vector equally inclined to the axes OX,

Question 12.

Question 13.

Question 14. If a, b, care mutually perpendicular vectors of equal magnitudes, show that the vector a +b + c is equally inclined to a, b and c.

Question 15.

Question 16. If q is the angle between two vectors a and b, then a b >0 only when

Question 17. Let a and b be two unit vectors and q is the angle between them. Then a + b is a unit vector if

Question 18.

Question 19.

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