FIND THE ANGLE BETWEEN TWO VECTORS

1. The vectors A and B are given by A = 3i + 5j − 4k and B = 8i − 4j + k. The angle between the two vectors is ......

   A. 90o	D. 37o
   B. 60o	E. 30o
   C. 45o

   
   
   Solution :
   To find the angle between two vectors, we can use the dot product formula. The dot product (the scalar product) of two vectors is defined by : 
   
   A.B = |A|.|B| cos θ
   
   With :
   |A| = the magnitude of vector A
   |B| = the magnitude of vector B
   θ = the angle between vectors A and B.
   
   According to the formula above :
   ⇒ A.B = |A|.|B| cos θ
   ⇒ A.B = |A|.|B| cos θ
   ⇒ (3i + 5j − 4k).(8i − 4j + k) = |A|.|B| cos θ
   ⇒ 3(8) + 5(-4) + (-4)(1) = |A|.|B| cos θ
   ⇒ 24 − 20 − 4 = |A|.|B| cos θ
   ⇒ 0 = |A|.|B| cos θ
   ⇒ cos θ = 0
   ⇒ θ = 90^o
   

   Answer : A

   
   
2. The angle between u = i + √2 j + √5 k and v = i − √2 j + √5  k is .....

   A. 90o	D. 30o
   B. 60o	E. 15o
   C. 45o

   
   
   Solution :
   According to the dot product formula :
   ⇒ u.v = |u|.|v| cos θ
   ⇒ u.v = |u|.|v| cos θ
   ⇒ (i + √2 j + √5 k).(i − √2 j + √5) = |u|.|v| cos θ
   ⇒ 1 − 2 + 5 = √1 + 2 + 5.√1 + 2 + 5 cos θ
   ⇒ 4 = 8 cos θ
   ⇒ cos θ = ½
   ⇒ θ = 60^o.
   

   Answer : B

   
   
3. The vectors A and B are given by A = 2i − j − 3k, and B = pi + j − k. If the two vectors are perpendicular to each other, the value of p is .....

   A. -2	D. 2
   B. -1	E. 3
   C. 1

   
   
   Solution :
   Since the two vectors are perpendicular to each other, so the angle between them is 90^o. According to scalar product formula :
   ⇒ a.b = |a|.|b| cos θ
   ⇒ a.b = |a|.|b| cos 90^o
   ⇒ (2i − j − 3k).(pi + j − k) = |a|.|b| (0)
   ⇒ 2(p) + (-1)(1) + (-3)(-1) = 0
   ⇒ 2p − 1 + 3 = 0
   ⇒ 2p + 2 = 0
   ⇒ 2p = -2
   ⇒ p = -1
   

   Answer : B