Show that the area of the triangle contained between the vector `overset rarr(a ) and overset rarr(b)` is one half of the magnitude of `overset rarr(a) xx overset rarr(b)`

Explanation

Let `vec(a)` be represented `vec(OP) " and" vec(b)` be represented by `vec(OQ)`. Let `anglePOQ=theta`, Fig. Complete the ||gm OPRQ. Join PQ.
Draw `QN bot OP`
IN `DeltaOQN, " " sin theta=(QN)/(OQ)=(QN)/(b)`
`QN=b sin theta`
Now, by definition , `|vec(a)xxvec(b)|=ab sin theta=(OP)(QN)`
`=(2(OP)(QN))/(2)=2xx " area of " DeltaOPQ`
`therefore " area of " DeltaOPQ=(1)/(2)|vec(a)xxvec(b)|`, which was to be proved.