The plane containing the line x-32=y+2-1=z-13 and also containing its projection on the pl…

Question

The plane containing the line $\dfrac{x-3}{2}=\dfrac{y+2}{-1}=\dfrac{z-1}{3}$ and also containing its projection on the plane $2x+3y-z=5$, contains which one of the following points?

Accepted Answer

The normal vector of required plane

$$ = \left( {2\widehat i - \widehat j + 3\widehat k} \right) \times \left( {2\widehat i + 3\widehat j - \widehat k} \right)$$

$$ =  - 8\widehat i + 8\widehat j + 8\widehat k$$

So, direction ratio of normal is $$\left( { - 1,1,1} \right)$$

So required plane is

$$ - \left( {x - 3} \right) + \left( {y + 2} \right) + \left( {z - 1} \right) = 0$$

$$ \Rightarrow  - x + y + z + 4 = 0$$

Which is satisfied by $$\left( {2,0, - 2} \right)$$

Solutions · 1