To calculate the distance between point $A = (1,2,3)$ and line $l = (1+t, 1-t, 5+2t)$.

Question

Calculate the distance between point $A = (1,2,3)$ and line $l = (1+t, 1-t, 5+2t)$.
I'm trying to find a point on the line l that is the closest distance to the point $A$ and then I can use Euclidean distance formula for two points in $3D$. But I'm having trouble finding the point on the line $l$ closest to the point $(1,2,3)$. Is there some formula I should use?

Answer 1

Distance of a Point from a Line Given in Component Form

For the distance $d$ of the point $M = (a, b, c)$ from a line given in the component form, i.e. $\frac{x-x_1}{l}=\frac{y-y_1}{m}=\frac{z-z_1}{n}$, we have: $$d^2=\frac{\left[(a-x_1)m-(b-y_1)l\right]^2+\left[(b-y_1)n-(c-z_1)m \right]^2+\left[(c-z_1)l-(a-x_1)n\right]^2}{l^2+m^2+n^2}.$$