Question

Show that the area of the triangle contained between the vectors a and b is one half of the magnitude of a × b.

Answer 1

Consider two vectors $OK→ = |a→| and OM→ = |b→|,$ inclined at an angle θ, as shown in the following figure.

In ΔOMN, we can write the relation:

$sinθ =\frac{ MN }{ OM }=\frac{ MN }{ |b→|}$
MN =$|\vec b|sinθ$
$|\vec a× \vec a| = |\vec a| |\vec b| sinθ$

= OK . MN × $\frac{2 }{ 2}$

= 2 × Area of ΔOMK

∴ Area of ΔOMK =\frac{ 1 }{ 2}$$ |\vec a ×\vec b|