Discussion
The unit vector that bisects the angle between two vectors \(2i + j + 2k\) and \(i + 2j - 2k\) is
Answer: E
The bisecting unit vector is \(\frac{i + j}{\sqrt{2}}\).
The internal bisector of two vectors lies along the sum of their unit vectors. Both given vectors have length 3, since \(\sqrt{4 + 1 + 4} = \sqrt{1 + 4 + 4} = 3\):
\[ \begin{aligned} \frac{2i + j + 2k}{3} + \frac{i + 2j - 2k}{3} &= \frac{3i + 3j}{3} \\ &= i + j \end{aligned} \]
Normalising, \(\frac{i + j}{\sqrt{2}}\). The difference of the unit vectors, \(\frac{i - j + 4k}{3}\), gives the external bisector, and none of the \(\sqrt{3}\) options is parallel to i + j.
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