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Question
If a unit vector `veca` makes an angles `pi/3` with `hati, pi/4` with `hatj` and an acute angle θ with `hatk`, then find θ and, hence the compounds of `veca`.
Solution
`veca = a_1hati + a_2hatj + a_3hatk, |veca| = 1`
`cos pi/3 = a_1/|veca|`
`1/2 = a_1[|veca| = 1]`
`cos pi/4 = a_2/|veca|`
`1/sqrt2 = a_2[|veca| = 1]`
`cos theta = a_3/|veca|`
`a_3 = costheta`
|a| = 1
`sqrt(a_1^2 + a_2^2 + a_3^2) = 1`
`(1/2)^2 + (1/sqrt2)^2 + cos^2theta = 1`
`1/4 + 1/2 + cos^2theta = 1`
`cos^2theta = 1/4`
`costheta = 1/2`
`theta = pi/3`
`a_3 = cos pi/3 = 1/2`
∴ `(1/2, 1/sqrt2, 1/2)`
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