Let O be the origin and the position vectors of A and B be 2 i+2 j+ k and 2 i+4 j+4 k resp…

Question

Let $O$ be the origin and the position vectors of $A$ and $B$ be $2\hat i+2\hat j+\hat k$ and $2\hat i+4\hat j+4\hat k$ respectively. If the internal bisector of $\angle AOB$ meets the line $AB$ at $C$, then the length of $OC$ is:

Accepted Answer

$|OA|=3,\ |OB|=6$. By the angle bisector, $C$ divides $AB$ such that $AC:CB=OA:OB=1:2$:
$$C=\frac{2A+B}{3}=\frac{(4,4,2)+(2,4,4)}{3}=\left(2,\frac83,2\right).$$
$$OC=\sqrt{4+\frac{64}{9}+4}=\sqrt{\frac{136}{9}}=\frac{2\sqrt{34}}{3}.$$

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