Q. The value of $\alpha$ for which point $M(\alpha \hat{i}+2 \hat{j}+\hat{k})$, lies in the plane containing three points $A(\hat{i}+\hat{j}+\hat{k})$, $B (2 \hat{ i }+2 \hat{ j }+\hat{ k })$ and $C (3 \hat{ i }-\hat{ k })$ is
Vector Algebra
Solution:
The equation of plane containing $A , B$ and $C$ is
$\begin{vmatrix}x -1 & y -1 & z -1 \\1 & 1 & 0 \\2 & -1 & -2\end{vmatrix}=0 $
$\Rightarrow -2 x +2 y -3 z +3=0$....(1)
$\text { Clearly, } M (\alpha, 2,1) \text { satisfy equation }(1) \text {, so }$
$-2 \alpha+4-3+3=0 \Rightarrow \alpha=2 $
