With respect to the origin $O$, the points $A$ and $B$ have position vectors given by $\overrightarrow{OA} = i + 2j + 2k$ and $\overrightarrow{OB} = 3i + 4j$. The point $P$ lies on the line $AB$ and $OP$ is perpendicular to $AB$.
$i)$ Find a vector equation of the line AB.
$ii)$ Find the position vector of $P$.
$iii)$ Find the equation of the plane which contains AB and which is perpendicular to the plane OAB, giving your answer in the form $ax + by + cz = d$.
I did part $i)$ and got $r = i + 2j + 2k +\lambda(2i + 2j – 2k)$
How can I do part $ii)$, I think once I do part $ii)$ I will be able to do $iii)$, any help please?
Best Answer
Let $\mathbf{OP} = <a,b,c> $.
We know that $AB$ and $\mathbf{OP}$ must be perpendicular, so the dot product of $\mathbf{OP}$ and $<2,2,-2>$ is $0$ ($<2,2,-2>$ plays the role of $\mathbf{v}$ in the diagram below)
$$<2,2,-2> \bullet <a,b,c> = 2a+2b-2c = 0$$ $$ \Rightarrow a+b-c=0$$ We also have that for some value of $\lambda$ (let's just call it $t$) $$<1,2,2> + <2,2,-2>t $$ $$=<1+2t, 2+2t, 2-2t> = <a,b,c>$$ because $P$ lies in the $AB $
Combining the two equations, we get $$1+2t +2 +2t - 2 +2t = 0 \Rightarrow 6t = -1$$ $$\Rightarrow t = -1/6$$
So $\mathbf{OP} = <2/3, 5/3, 7/3> = \frac13 <2,5,7>$