The vector a= i+2 j+ k lies in the plane of the vectors b= i+ j and c= j+ k and bisects th…

Question

The vector $\vec a=\alpha\hat i+2\hat j+\beta\hat k$ lies in the plane of the vectors $\vec b=\hat i+\hat j$ and $\vec c=\hat j+\hat k$ and bisects the angle between $\vec b$ and $\vec c$. Then which one of the following gives possible values of $\alpha$ and $\beta$?

Accepted Answer

Since $|\vec b|=|\vec c|=\sqrt2$, the angle bisector direction is $\vec b+\vec c=\hat i+2\hat j+\hat k$. So $\vec a=k(\hat i+2\hat j+\hat k)$ for some scalar $k$. Comparing the $\hat j$ coefficient: $2=2k \implies k=1$, giving $\alpha=1,\beta=1$.

Solutions · 1