If a unit vector ` vec a` makes angles `pi/3` with ` hat i ,pi/4` with ` hat j` and an acute angle `theta` with ` hat k` , then find the value of `theta` .

Explanation

If a unit vector makes an angle `alpha` with `x`-axis, `beta` with `y`-axis and `gamma` with `z`-axis, then,

`cos^2 alpha + cos^2 beta +cos^2 gamma = 1`

Here, `alpha = pi/3, beta = pi/4 and gamma = theta`

`:. cos^2 (pi/3) + cos^2(pi/4) +cos^2theta = 1`

`=>1/4+1/2+cos^2theta = 1`

`=>cos^2 theta = 1-3/4 = 1/4`

`=>cos theta = +-1/2 = cos(pi/3)` (Given that angle is acute)

`:. theta = pi/3`