VECTOR ALGEBRA. 3. write a linear vector in the direction of the sum of the vector a = 2i + 2j 5k and

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1 1 mark questions VECTOR ALGEBRA 1. Find a vector in the direction of vector 2i 3j + 6k which has magnitude 21 units Ans. 6i-9j+18k 2. Find a vector a of magnitude 5 2, making an angle of π with X- axis, π 4 2 acute angle θ with Z- axis with Y- axis and an Ans. 5i + 5k 3. write a linear vector in the direction of the sum of the vector a = 2i + 2j 5k and b = 2i + j 7k Ans. 4i+3j 12k Find the value of p for which the vectors 3i + 2j + 9k and i 2pj 3k are parallel Ans. (-1/3) 5. Write the value of cosine of the angle which the vector a = i + j + k makes with Y-axis Ans find the angle between X-axis and the vector i + j + k Ans. cos write a vector in the direction of the vector i 2j + 2k that has magnitude 9 units Ans. 3i-6j+6k 8. write a unit vector in the direction of vector PQ, where P and Q are the points (1,3,0) and (4,5,6) respectively Ans. 3i 2j+6k 7 9. write the value of the following i (j + k ) + i (j + k ) + i (j + k ) Ans. 0

2 10. write a unit vector in the direction of the sum of vectors a = 2i j + 2k and b = i + j + 3k Ans. i+5k if a = xi + 2j zk and b = 3i yj + k are two equal vectors, then write the value of x + y + z Ans P and Q are two points with position vectors 3a 2b and a + b, respectively. write the position vector of a point R which divides the line segment PQ in the ratio 1 2 externally Ans. a+ 4b 13. L and M are two points with position vectors 2a b and a + 2b, respectively. write the position vector of a point N which divides the line segment LMin the ratio 1 2 externally Ans. 5b 14. A and B are two points with position vectors 2a 3b and 6b a, respectively. write the 2 1 position vector of a point P which divides the line segment AB internally in the ratio Ans. a 15. Find the sum of the vectors a = i 2j + k, b = 2i + 4j + 5k and c = i 6j 7k. Ans. -4j - k 16. find the scalar components of AB with initial pointsa(2,1) and terminal point B( 5,7) Ans for what value of a, the vectors 2i 3j + 4k and ai + 6j 8k are collinear Ans. (-4) 18. write the cosines of vector 2i + j 5k Ans. 2 30, 1, write the position vector of the mid-point of the vector joining points P(2,3,4) and Q(4,1, 2) Ans. 3i+2j+k

3 20. write a unit vector in the direction of vector a = 2i + j + 2k Ans. (2i+j+2k) find the magnitude of the vector a = 3i 2j + 6k. Ans. 7 Units 22. Find a unit vector in the direction of vector a = 2i + 3j +6k. Ans. (2i+3j+6k) If A, B and C are the vertices of a ABC, then what is the value of AB + BC + CA? Ans. 4i-2j+4k 24. Find a unit vector in the direction of a =2i-3j+6k. Ans. (2i 3j+6k) Find a vector in the direction a =2i j +2k which has magnitude 6 units. Ans Find a position vector of mid point of the line segment AB,where A is point (3,4, -2)and B is point (1,2,4). Ans. 2i-3j+k 27. Write a vector of magnitude 9 units in the direction o9f vector -2i + j + 2k. Ans. -6i-3j+6k 28. Write a vector of magnitude 15 units in the direction of vector i 2j + 2k. Ans. 5i-10j+10k 29. What is the cosine of angle which the vector 2i + j + k makes with y-axis? Ans. 1/2 30. Find a units vector in the direction of vector b = 6i -2j + 3k.

4 Ans. (6i 2j+3k) Find a units vector in the direction of vector a = -2i + j + 2k. Ans. ( 2i+j+2k) Write a unit vector in the direction of a = 2i 6j 3k. Ans. (2i 6j 3k) Find the magnitude of the vector a = 2i 6j 3k. Ans. 7 umits 34. Find a unit vector in the direction of a = i + j + 2k. Ans. (i+j+2k) If a = i + 2j k and b = 3i + i 5k,then find a unit vector in the direction a b. Ans. ( 2i+j+4k) If a and b are perpendicular vectors, a +b =13 and b =5, then find the value of b. Ans If a and b are two unit vectors such that a + b is also a unit vector, then find the angle between a and b. Ans. 2π/3 38. Find the projection of the vector i + 3j + 7k on the vector 2i 3j + 6k. Ans Write the projection of vector i + j +k along the vector j. Ans If vectors a and b are such that a = 3, b =2/3 and a * b is a unit vector then write the Angle between a and b.

5 Ans. π/6 41. Find a (b * c ), if a = 2i + j + 3k, b = -I + 2j + k and c = 3j + j + 2k. Ans If a and b are unit vectors, then find the angle between a and b, given that( 3a b is a unit vector Ans. π/6 43. If a = 8, b = 3 and a b =12, find the angle between a and b Ans. π/6 44. Write the projection of the vector a = 2i j + k on the vector b = i +2j + 2k Ans. 2/3 45. Write the value of x so that the vectors a = 2i + xj + k and b = i 4j + 3k are perpendicular to each other. Ans. 5/2 46. Write the projection of the vector7i + j 4k on the other vector 2i + 6j + 3k. Ans. 8/7 47. If a and b are two vectors such that a + b = a, then prove that vector 2 a + b is Perpendicular to vector b 48. Find x, if for a unit vector a (x a). (x + a) =15. Ans Find x when projection of a =xi + j + 4k on b = 2i + 6j + 3k is 4 units Ans Write the value of (k j ).i +j +k. Ans. (-1)

6 51. If a. a= 0 and a. b =0, then what can be concluded about the vector b? Ans. b is either zero or non-zero perpendicular vector. 52. Write the projection of vector i j on the vector i + j. Ans Write the angle between vectors a and b with magnitudes 3 and 2 respectively, having a. b = 6. Ans. π/4 54. For what value of x are the vectors i + 2xi + j 3k perpendicular. Ans. 1/2 55.If a = 3, b = 2and angle between a and b is 60, then find a. b Ans find the value of x, if the vectors 2i + xi + 3k and 3i + 2j -4k are perpendicular to each other. Ans If a = 2, b = 3 and a. b = 3, then find the projection of b on a. Ans. 3/2 58. Vectors a = 3, b =2/3 and a*b is a unit vector. Write the angle between aand b. Ans. π/3 59. If aand b are two vectors, such that a. b = a*b, then find the angle between a*b Ans. π/4 60. Find λ, if (2i+6j+14k)*(i-λj+7k)=0.

7 Ans. (-3) 61. Find a. b, if a =-i+j-2k and b =2i+3j-k. Ans Find a. b, if a=3i-j+2kand b =2i+3j+3k. Ans Find the value of P, if (2i+6j+27k)*(i+3j+Pk)=0. Ans. 27/2 64. If P is a unit vector and (x-p).(x+p)=80, then find x. Ans Find the angle between a and b with magnitude 1 and 2 respectively, when a*b = 3. Ans. π/3 66. Write the value of P for which a=3i+2j+9k and b=i+pj+3k are parallel vectors. Ans. 2/3 67. Find the projection of a on b, if a. b =8 and b =2i+6j+3k. Ans. 8/7 68. Find value of the following: i.(j*k)+j(i*k)+k(i*j). Ans. 1

8 69. Find a*b, if v=i-7j+7k and b=3i-2j+2k. Ans If a = 3, b =2 and a. b =3, then find the angle between a and b. Ans. π/6 71. Find angle between vectors a=i-j+k and b=i+j-k. Ans. cos Marks Questions

9 1. Find a vector of magnitude 5 units and parllel to the resultant of a =2i +3j k and b= i 2j +k to the vector 2 a b + 3 c. Ans. (15i+5j) Find the position vector of a point R, which divides the line joining two points P and Q whose position vectors are 2 a + b and a 3 b respectively externally in the ratio 1:2. Also, show that P is the mid- point of line segment RQ. Ans. 3a +5b 3. Prove that for any three vectors a, b and c, [a+b b +c c+a]=2[a b c]. 4. Vectors a, b and c are such that a+b +c =0 and a =3, b =5, c =7. Find the angle between a and b. Ans. π/3 5. Show that the four points A, B, C and D with position vectors 4i+5j+k,-j-k, 3i+9j+4k and 4(-i+j+k) respectively are coplanar. 6. The scalar product of the vectors a=i +j +k with the unit vector along the sum of the vectors b =2i+4j-5k and c =λi+2j+3k is equal to one. Find the value of λ and hence, find the unit vector along b +c. Ans Find the vector p which is perpendicular to both α=4i+5j-k and β=i-4j+5k and p. q =21, where q =3i+j-k. Ans. 7i-7j-7k 8. Find a unit vector perpendicular to both of the vectors a+b and a -b where a=i+j+k, and b =i+2j+3k.

10 Ans. ( i+2j k) 6 43) Find the unit vector perpendicular to the plane ABC where the position vectors of A, B and C are 2i-j+k, i+j+2k and 2i+3k respectively. Ans. (3i+2j k) Show that the vectors a, b and c are coplanar if and only if a +b, b + c and c+a are coplanar. 45. If a, b and c are three mutually perpendicular vectors of the same magnitude then prove that a+ b + c is equally inclined with the vectors a, b and c If a =i + j + k and b = j k, then find a vector c, such that a c = b and a. c = 3. Ans. 9i+7j+12k 47. Using vectors, find the area of the ABC, whose vertices are A(1, 2, 3), B(2, -1, 4) and C ( 4, 5, -1). Ans If a = i j + 7k and b = 5i j + δk, then find the value of δ, so that a+ b and a b are perpendicular vectors. Ans. ±5 12. If p = 5i + δ j 3k and q = i + 3j 5k, then find the value of δ, so that p + q and p q are perpendicular vectors. Ans. ±1 13. If a, b and c are three vectors, such that a = 5, b = 12, c = 13 and a + b + c = 0, then find the value of a.b +b.c + c.a. Ans. (-169)

11 14. Let a = i + 4j +2k, b = 3i 2j + 7k and c = 2i j + 4k. Find a vector p, which is perpendicular to both a and b and p.c = 18. Ans. 64i -2j -28k 15. Find a unit vector perpendicular to each of the vectors a + b and a b, where a = 3i + 2j + 2k and b = i + 2j 3k. Ans. (2i 2j k) If a and b are two vectors, such that a = 2, b = 1 and a.b = 1, then find (3a 5b ).(2a+7b ). Ans If vectors a = 2i +2j +3k, b = -i + 2j + k and c = 3i + j are such that a + δ b is perpendicular to c, then find the value of δ. Ans Using vectors, find the area of triangle with vertices A(1,1,2), B(2,3,5) and C(1,5,5). Ans If a, b and c are three vectors, such that a = 3, b = 4 and c = 5 and each one of these are perpendicular to the sum of other two, then find a+b +c. Ans Using vectors, find the area of the triangle with vertices A(2,3,5),B(3,5,8) and C(2,7,8). Ans The scalar product of vector i +j + k with the unit vector along the sum of vectors 2i + 4j 5k and δ i + 2j + 3k is equal to one. Find the value of. Ans If a b = c d and a c = b d, then show that a d is parallel to b c, where a d and b c. Ans. 0

12 23. Three vectors a, b and c satisfy the condition a + b + c = 0. Find the value of a.b + b.c + c.a, if a = 1, b = 4 and c = 2. Ans Find a vector of magnitude 5 units, perpendicular to each of the vectors (a + b ) and (a b ), where a = i + j + k and b = i +2j +3k. Ans. (5i+10j 5k) 6

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