DPP
DPP
DPP
M
P Q
a
Now, area of parallelogram = | PQ | · | SM | = | PQ · | PS | sin = a |·| b | sin = | a × b | hence cross
product of two vectors represents the area of parallelogram formed by it. It is worth noting that area
vector a × b acts along the perpendicular to the plane of two vectors a and b .
Q-1. Find a × b and b × a if
(i) a = 3 k̂ + 4 ˆj , b = î + ˆj – k̂
(ii) a = (2, –1, 1) ; b = (3, 4, –1)
Q-2. If a = 3 î + ˆj + 2 k̂ and b = 2 î – 2 ˆj + 4 k̂
(i) find the magnitude of a × b
(ii) find a unit vector perpendicular to both a and b.
(iii) find the cosine and sine of the angle between the vectors a and b
Q-3. The vectors from origin to the points A and B are a = 3iˆ 6ˆj 2kˆ and b = 2iˆ ˆj 2kˆ respectively..
Find the area of :
(i) the triangle OAB
(ii) the parallelogram formed by OA and OB as adjacent sides.
VECTOR PRODUCT
1. If F 2iˆ 3 ˆj kˆ and r iˆ ˆj 6kˆ find r F
(1) 17i 13 j 5k (2) 17i 13 j 5k (3) 3i 4 j 5k (4) 3i 4 j 5k
2. Two sides of a triangle are given by i j k and i 2j 3k , then area of triangle is
4.
Area of a parallelogram formed by vectors 3iˆ - 2jˆ + kˆ m and ˆi + 2jˆ + 3kˆ m as adjacent sides is
2 36 5 15 23 36 15
(1) x 0, y (2) x ,y (3) x ,y (4) x ,y
3 5 3 3 5 3 14
6. A B B A is equal to
(1) 2 AB (2) A2 B2 (3) zero (4) null vector
7. If A and B are two vectors, then which of the following is wrong ?
(1) A.B B. A (2) A B B A (3) A B B A (4) A B B A
8. If none of the vectors A, B and C are zero and if AB 0 and BC 0 the value of AC is
(1) Unity (2) Zero (3) B2 (4) AC cos
9. If A , B and C are coplanar vectors, then
(1) A.B C 0
(2) A B .C 0
(3) A.B .C 0 (4) all the above are true
10. If A along North and B along vertically upward the direction of A B is along
(1) west (2) south (3) east (4) vertically downwards
ANSWER KEY
1. 3 2. 4 3. 3 4. 2 5. 4
6. 1 7. 4 8. 3 9. 1 10. 2
ANSWER KEY
1. 1 2. 2 3. 1 4. 3 5. 2
6. 4 7. 3 8. 2 9. 2 10. 3