M denotes the mid-point of the line segment joining A$(4\hat{i} + 5\hat{j}- 10 \hat{k})$ and $B(-\hat{i} + 2\hat{j} + \hat{k}),$ then the equation of the plane through M and perpendicular to AB, is
Answer & explanation
Correct answer: option 1
The position vector of M is $\frac{3}{2}\hat{i}+\frac{7}{2}\hat{j}-\frac{9}{2}\hat{k}$
$\vec{AB}= - 5\hat{i} - 3\hat{j} + 11\hat{k}$
The equation of the plane through M, perpendicular to $\vec{AB}$ is
$\begin{Bmatrix} \vec{r}- \left(\frac{3}{2}\hat{i}+\frac{7}{2}\hat{j}-\frac{9}{2}\hat{k}\right)\end{Bmatrix}. (-5\hat{i}- 3\hat{j}+11\hat{k}) = 0 $
$⇒\vec{r}.(-5\hat{i}- 3\hat{j}+11\hat{k}) +\frac{135}{2}= 0 $