Question

Find the component form of `bar"a"` if it lies in YZ-plane makes 60° with positive Y-axis and `|bar"a"| = 4`.

Answer 1

Let α, β, γ be the direction angles of `bar"a"`

Since `bar"a"` lies in YZ-plane, , it is perpendicular to X-axis

∴ α = 90°

It is given that β = 60°

∵ cos²α + cos²β + cos²γ = 1

∴ cos²90° + cos²60° + cos²γ = 1

∴ 0 + `(1/2)^2` + cos²γ = 1

∴ cos²γ  = `1 - 1/4 = 3/4`

∴ cos²γ = `+- sqrt3/2`

Unit vector along `bar"a"` is given by

`hat"a" = ("cos" alpha)hat"i" + ("cos"beta)hat"j" + ("cos"gamma)hat"k"`

`= 0.hat"i" + 1/2hat"j" + sqrt3/2hat"k"`

`= 1/2hat"j" +- sqrt3/2hat"k"`

∴ `bar"a" = |bar"a"|hat"a" = 4(1/2hat"j" +- sqrt3/2hat"k")`     .....[∵ `|bar"a"| = 4`]

∴ `bar"a" = 2hat"j" +- 2sqrt3hat"k"`