Question

Find the perpendicular distance of the vertex A from the base of a $\Delta ABC$ with a,b,c as position vectors of A,B,C respectively.

• A. $p=\frac{|a\times b-b\times c-c\times a|}{|b-c|}$
• B. $p=\frac{|a\times b+b\times c+c\times a|}{|b-c|}$
• C. $p=\frac{|a\times b-b\times c-c\times a|}{|a-b|}$
• D. $p=\frac{|a\times b+b\times c+c\times a|}{|a-b|}$

Answer 1

The correct option is B $p=\frac{|a\times b+b\times c+c\times a|}{|b-c|}$

Area of $\Delta ABC$ $=\frac{1}{2}|a\times b+b\times c+c\times a|=\frac{1}{2}BC.p$
and $BC=|b-c|.$
Hence length of perpendicular, from A on BC
$i.e.p=\frac{|a\times b+b\times c+c\times a|}{|b-c|}$