Q.If a line drawn from the point $A(1, 2, 1)$ is perpendicular to the line joining $P(1, 4, 6)$ and $Q(5, 4, 4)$ then find the co-ordinates of the foot of the perpendicular.

The perpendicular distance of a point $P(\bar{\alpha})$ from a line $\bar{r} = \bar{a}+\lambda\bar{b}$ can be found without ever locating the actual foot of the perpendicular, by combining a projection with the Pythagorean theorem. If $M$ is the foot of the perpendicular from $P$ onto the line, then $AM$ — the distance along the line from the line's base point $A$ to $M$ — is exactly the scalar projection of $\overline{AP} = \bar{\alpha}-\bar{a}$ onto the line's direction $\bar{b}$, namely $AM = \dfrac{(\bar{\alpha}-\bar{a})\cdot\bar{b}}{|\bar{b}|}$. Since triangle $AMP$ is right-angled at $M$, Pythagoras' theorem gives $PM^2 = AP^2 - AM^2$, so the required perpendicular distance is $PM = \sqrt{|\bar{\alpha}-\bar{a}|^2 - \left[\frac{(\bar{\alpha}-\bar{a})\cdot\bar{b}}{|\bar{b}|}\right]^2}.$ This single formula applies directly whenever the line is already given in vector form. When a line is instead given in Cartesian symmetric form, it is often just as quick, and sometimes q …