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 =\(\frac{ 1 }{ 2}\)\( |\vec a ×\vec b|\)