Question

If a+2b and 5a-4b are perpendicular to each other and a and b are unit vector. Find the angle between a and b​

Answer 1

Step-by-step explanation:

We are given that A and B are two unit vectors ie | A | = 1 and | B | = 1.

Let ß be the angle between the vectors A and B, such that 0° ≤ ß ≤ 180°.

We are given that (A + 2 B) and (5 A - 4 B) are perpendicular to each other. This means that the dot product between (A + 2 B) and (5 A - 4:B) is zero, ie

(A + 2 B) . (5 A - 4 B) = 0

=> 5 A . A + 10 B . A - 4 A . B - 8 B . B = 0

=>5 | A |² + 10 A . B - 4 A . B - 8 B . B = 0 —(1)

Now

| A |² = 1; | B | = 1; and B . A = A . B (dot product of two vectors commutes), so eqn 1 becomes:

=> 5 × 1 + 6 A . B - 8 × 1 = 0

=> 6 | A | × | B | Cos ß - 3 = 0

==> 6 × 1 × 1 Cos ß - 3 = 0

=> Cos ß = 3/6 = 1/2

ß = arc Cos ( 1/2 ) = 60°

The angle between vectors A and B is 60°.