above the paper while ˆ n directed below the paper. i.e. ˆ n n = − Fig . Hence .
In view of the Observations and , we have × = − = − = − . If represent the adjacent sides of a triangle then its area is given as By definition of the area of a triangle, we have from Fig . , Area of triangle ABC = AB CD. But (as given), and CD = sin θ .
Thus, Area of triangle ABC = . If represent the adjacent sides of a parallelogram, then its area is given by From Fig . , we have Area of parallelogram ABCD = AB. DE.
But (as given), and Thus, Area of parallelogram ABCD = We now state two important properties of vector product. Property (Distributivity of vector product over addition): If are any three vectors and λ be a scalar, then (i) (ii) Fig . Fig . Let be two vectors given in component form as a i a j a k b i b j b k , respectively.
Then their cross product may be given by Explanation We have a i a j a k b i b j b k a b i a b i a b i a b k a b k a b k (by Property ) a b i a b k a b i a b k ˆ ˆ (as and × = = −× × = −× = −× a b k a b j a b k a b i a b j a b i (as × = ˆ a b i a b k Example Find Solution We have ( ) ( )