How do you derive the formula $d= \frac {|a \times b| }{|a|}$ to find the shortest distance between 2 vectors?

Question

Distance from point $P$ (not on $L$) to line $L$ (that passes through $Q$ and $R$) is $$d=\frac{|\vec{a}\times \vec{b}|}{|\vec{a}|}$$

where $\vec{a}=\vec{QR}$ and $\vec{b}=\vec{QP}$

Find the distance from the given point to the given line:

(a) $(4, 1, −2); x = 1 + t, y = 3 − 2t, z = 4 − 3t$

How do you derive the above formula to find the shortest distance between the point and the vector?

Answer 1

Geometrically, the formula is saying $PH$ equals the area of the parallelogram divided by $QR$.

Answer 2

$d= \frac{|a \times b|}{|a|}=|bsin(\alpha)| $ where $\alpha$ is the angle between the vectors $a$ and $b$.