Let R and a= i+ j+3 k, b=2 i+ j- k, c= i-2 j+3 k.

Question

Let $\alpha\in\mathbb R$ and $\vec a=\alpha\hat i+\hat j+3\hat k$, $\vec b=2\hat i+\hat j-\alpha\hat k$, $\vec c=\alpha\hat i-2\hat j+3\hat k$. The set $S=\{\alpha:\vec a,\vec b,\vec c	ext{ coplanar}\}$:

Accepted Answer

$$[\vec a\ \vec b\ \vec c] = \begin{vmatrix}\alpha&1&3\2&1&-\alpha\ \alpha&-2&3\end{vmatrix}=0$$

Expanding gives $\alpha^2+6=0$, which has no real solution. So $S=\varnothing$.

Solutions · 1