Calculus III Vector distance problem.

Question

Here is the question:

The distance,d, of a point P to the line through points A and B is the length of the component of AP that is orthogonal to AB, as indicated in the diagram below.

So the distance from $P=(−4,−5,−1)$ to the line through the points $A=(1,2,−4)$ and $B=(5,−2,−5)$ is_____?

The diagram is:

If someone could explain to me how to do this without giving the answer away, that would be much appreciated. I have tried this problem three times and all three answers are wrong. For all three of my attempts, I used the equation: $\dfrac{a\cdot{b}}{|a|}$, where the top is a dot product and then divided by the unit vector of a. I am not sure if this is the wrong equation or if I am approaching the problem incorrectly.

Answer 1

Hint: $\;\vec{AP}=\dfrac{\vec{AP} \cdot \vec{AB}}{\left|\vec{AB}\right|^2} \, \vec{AB} + \vec{d}\,$, so $\vec{d} = \vec{AP} - \dfrac{\vec{AP} \cdot \vec{AB}}{\left|\vec{AB}\right|^2} \, \vec{AB}\,$ where the RHS is easily calculated.

Answer 2

$\frac{a \cdot b}{|a|}$ is not the formula for the perpendicular component, but for the magnitude of the parallel component. However, once you know the length of the parallel projection, you can then find the parallel projection. Can you then find a relationship between $AP$, $proj_{AB}AP$, and $orth_{AB}AP$?