# Find the vector equation of the plane through the points (2,1,-1) and (-1,3,4) and perpendicular to the plane x-2y+4z=10.

(Note: This question has been split into 2 questions)

This question appeared in 65-1,65-2 and 65-3 versions of the paper in 2013.

$\textbf{Step 1}$:
The required plane passes through: $P(2,1,-1)$ and $Q(-1,3,4)$.
Then we can write the position vectors as follows: $a_1=2\overrightarrow{i}+\overrightarrow{j}-\overrightarrow{k}$ and $a_2=-\overrightarrow{i}+3\overrightarrow{j}+4\overrightarrow{k}$
$\Rightarrow PQ=(a_2-a_1)=-3\overrightarrow{i}+2\overrightarrow{j}+5\overrightarrow{k}$
$\textbf{Step 2}$:
Let $\overrightarrow{i}$ be the normal vector to the desired plane $\overrightarrow{n}=\overrightarrow{n_i} \times \overrightarrow{PQ}$ =$\begin{vmatrix} \overrightarrow{i} & \overrightarrow{j} & \overrightarrow{k}\\ 1 & -2 & 4 \\ -3 & 2 & 5 \end{vmatrix}$
$\Rightarrow \overrightarrow n=\overrightarrow{i}(-10-8)-\overrightarrow{j}(5+12)+\overrightarrow{k}(2-6) = -18\overrightarrow{i}-17 \overrightarrow{j}-4 \overrightarrow{k}$
$\textbf{Step 3}$:
$\Rightarrow \overrightarrow{r}.\overrightarrow{n} =\overrightarrow{a}.\overrightarrow{n}$
$\Rightarrow \overrightarrow r.(-18\overrightarrow{i}-17\overrightarrow{j}-4\overrightarrow{k}) = (2\overrightarrow{i}+\overrightarrow{j}-\overrightarrow{k})(-18\overrightarrow{i}-17\overrightarrow{j}-4\overrightarrow{k})$
$\Rightarrow$ $\overrightarrow{r}.(-18\overrightarrow{i}-17\overrightarrow{j}-4\overrightarrow{k})=-36-17+4$
$\Rightarrow$ $\overrightarrow{r}.(18\overrightarrow{i}-17\overrightarrow{j}+4\overrightarrow{k})=49$
edited Mar 22, 2013