Question

If `bar"a".bar"b" = sqrt3` and `bar"a" xx bar"b" = 2hat"i" + hat"j" + 2hat"k"`, find the angle between `bar"a"` and `bar"b"`.

Answer 1

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

∵ `bar"a" xx bar"b" = 2hat"i" + hat"j" + 2hat"k"`

∴ `|bar"a" xx bar"b"| = sqrt(2^2 + 1^2 + 2^2) = sqrt(4 + 1 + 4) = 3`

∴ `|bar"a"||bar"b"|` sin θ = 3      ...(1)

∴ `bar"a".bar"b" = sqrt3`

∴ `|bar"a"||bar"b"| "cos" theta = sqrt3`     ....(2)

∴ Dividing (1) by (2), we get

`(|bar"a"||bar"b"| "sin" theta)/(|bar"a"||bar"b"| "cos" theta) = 3/sqrt3`

∴ tan θ = `sqrt3 = tan 60^circ`

∴ θ = 60°