Distance of the point P(c) from the line r = a+λ b, is

We have,

QM = Projection of $\vec{QP}$ on $\vec{b}$

$⇒ QM = |\vec{QP}.\vec{b}|= |(\vec{c}-\vec{a}).\vec{b}| = \frac{|(\vec{c}-\vec{a}).\vec{b}|}{|\vec{b}|}$

 

In right angled triangle PMQ, we have

$PM^2 = PQ^2 - QM^2$

$⇒ PM = \sqrt{|\vec{c}-\vec{a}|^2=\frac{|(\vec{c}-\vec{a}).\vec{b}|^2}{|\vec{b}|^2}}$

$⇒ PM = \sqrt{\frac{|\vec{c} - \vec{a}|^2|\vec{b}|^2-|(\vec{c}-\vec{a}).\vec{b}|^2}{|\vec{b}|^2}}$

$⇒ PM = \frac{|(\vec{c}-\vec{a}).\vec{b}|}{|\vec{b}|}$    $[∵ |\vec{α}×\vec{β}|^2 = |\vec{α}|^2|\vec{β}|^2 - (\vec{α}.\vec{β})^2 ]$