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Question
Let `veca` and `vecb` be two unit vectors, and θ is the angle between them. Then `veca + vecb` is a unit vector if ______.
Options
`theta = pi/4`
`theta = pi/3`
`theta =pi/2`
`theta = (2pi)/3`
Solution
Let `veca` and `vecb` be two unit vectors, and θ is the angle between them. Then `veca + vecb` is a unit vector if `underline(theta = (2pi)/3)`.
Explanation:
Let `veca` and `vecb` be unit vectors and θ is the angle between them,
Then `|veca| = |vecb| = 1`
Now `veca + vecb` is a unit vector if `|veca + vecb| = 1`
`|veca + vecb| = 1`
`(veca + vecb)^2 = 1`
`(veca + vecb) xx (veca + vecb) = 1`
`veca xx veca + veca xx vecb + vecb xx veca + vecb xx vecb = 1`
`|a|^2 + 2a xx b + |b|^2 = 1`
`1^2 + 2|veca||vecb|costheta + 1^2 = 1`
`1 + 2.1.1costheta + 1 = 1`
`cos theta = (-1)/2`
`theta = (2pi)/3`
Hence, `veca + vecb` is a unit vector if `theta = (2pi)/3`
The correct answer is `(2pi)/3`.
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