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If a unit vectorvec(a) makes angle (pi)/...

If a unit vector`vec(a)` makes angle `(pi)/(3)` with `hat(i), (pi)/(4)` with `hat(j)` and `theta in ((pi)/(2), pi)` with `hat(k)` then a value of `theta` is `(k pi)/(l)` (k and l are coprime). The value of k is _________.

Text Solution

Verified by Experts

The correct Answer is:
5
`overline(a).hat(i) = cos (pi)/(3) = (1)/(2) = l`
`overline(a).hat(j) = cos (pi)/(4) = (1)/(sqrt(2)) = m`
Let `overline(a).hat(k) = n = cos theta`
we know that
`l^(2) + m^(2) + n^(2) = 1,` `(1)/(4) + (1)/(2) + n^(2) = 1`
`n = +- (1)/(2) = cos theta`, `theta = (pi)/(3), (2 pi)/(3)`
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