Question

Find `bar"u".bar"v"` if `|bar"u"| = 2, |bar"v"| = 5, |bar"u" xx bar"v"| = 8`

Answer 1

Let θ be the angle between `bar"u"` and `bar"v"`

Then `|bar"u" xx bar"v"| = 8` gives

`|bar"u"||bar"v"|` sin θ = 8

∴ 2 × 5 × sin θ = 8

∴ sin θ = `4/5`

cos θ = `+- sqrt(1 - "sin"^2theta)`    ...[∵ 0 ≤ θ ≤]

`= - sqrt(1 - (4/5)^2)`

`= +-sqrt(1 - 16/25)`

`= - sqrt(9/25) = +-3/5`

Now, `bar"u".bar"v" = |bar"u"||bar"v"|` cos θ

∴ `bar"u".bar"v" = 2 xx 5 xx (+- 3/5) = +- 6`