# How to find perpendicular distance from point to plane in $3D$.

The line $L_1$ passes through point $A$ whose position vector is $3i - 5j + 4k$, and is parallel to the vector $3i + 4j + 2k$. The line $L_2$ passes through the point $B$ whose position vector is $2i + 3j + 5k$, and is parallel to the vector $i - j - 4k$. The point $P$ on $L_1$ and $Q$ on $L_2$ are such that $PQ$ is perpendicular to both $L_1$ and $L_2$. The plane $\pi_1$ contains $PQ$ and $L_1$, and the plane $\pi_2$ contains $PQ$ and $L_2$.

(i) Find the length of PQ.
Since $PQ$ is perpendicular to both lines, the cross product of both lines' directions is the direction vector of $PQ$ $\Rightarrow (3i + 4j + 2k) \times (i - j - 4k) = -7(2i - 2j + k).$ Hence $|PQ| = \sqrt{(4+4+1)} = \sqrt{9} = 3.$

(ii) Find a vector perpendicular to π1.
Since $\pi_1$ contains $PQ$ and $L_1$, the cross product of both direction vectors will be the normal vector of $\pi_1$ $\Rightarrow (2i - 2j + k) \times (3i + 4j + 2k) = -(8i + j - 14k).$

(iii) Find the perpendicular distance from $B$ to $\pi_1$.
$B$ was given as the position vector $(2i + 3j + 5k)$, which lies on the line $L_2$.

I can't figure out how to get the equation of $\pi_1$ because it involves getting a point on the plane (which would be $P$), and I can't figure out how to get point $P$ either.

If I had point $P$, I would have found the equation of $\pi_1$, and then, using the distance formula, found the shortest distance between $B$ and $\pi_1.$

How can I find this distance? Please explain

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A piece of (unasked) advice: you're using physics notation, which for some of us poor mortal mathematicians is pretty annoying to read. Try to "translate" your notation to mathematical one (i.e., $\,8i+j-14k=(8,1,-14)\,$ and perhaps (just perhaps) more people around here will try harder to read all the above. – DonAntonio Feb 13 '13 at 11:31
@DonAntonio this is maths A level, we are required to use many notations including this one; I personally prefer column notation however this is easier to type. Why is it so hard to read?? – Ozzy Feb 13 '13 at 18:40