Question

Let `bar"a"` and `bar"b"` be two unit vectors. If the vectors `bar"c" = bar"a" + 2bar"b"` and `bar"d" =  5bar"a"  - 4bar"b"` are perpendicular to each other, then the angle between `bar"a"`  and `bar"b"` is ______.

Options

• `pi/6`

• `pi/2`

• `pi/3`

• `pi/4`

Answer 1

Let `bar"a"` and `bar"b"` be two unit vectors. If the vectors `bar"c" = bar"a" + 2bar"b"` and `bar"d" =  5bar"a"  - 4bar"b"` are perpendicular to each other, then the angle between `bar"a"`  and `bar"b"` is `pi/3`.

Explanation:

Let θ be the angle between `bar"a"` and `bar"b"`.

Sine, `bar"c" = bar"a" + 2bar"b"` and `bar"d" = 5bar"a" - bar"b"` are erpendicular to each other.

∴ `bar"c"*bar"d"` = 0

⇒ `(bar"a" + 2bar"b")*(5bar"a" - 4bar"b")` = 0

⇒ `5(bar"a"*bar"a") + 6(bar"a"*bar"b") - 8(bar"b"*bar"b")` = 0

⇒ `5|bar"a"|^2 + 6|bar"a"||bar"b"|costheta - 8|bar"b"|^2` = 0

⇒ `5 + 6 cos theta - 8` = 0

⇒ `cos theta = 1/2`

⇒ θ = `pi/3`