## Formula :

$$c\;=\;a*b\;=\;|a|\;*\;|b|\;*\;sinθ * n$$

Simplify with example:

$$\text{C}\;=\;\begin{vmatrix}i & j & k \\a_1 & a_2 & a_3 \\b_1 & b_2 & b_3 \\ \end{vmatrix}$$

$$c=[(a_2*b_3)-(a_3*b_2)]i+[(a_3*b_1)-(a_1*b_3)]j+[(a_1*b_2)-(a_2*b_1)]k$$

## Example :

Suppose that a parallelogram whose adjacent sides are defined by the two vectors a (6, 3, 1) and b (3, -1, 5).Calculate the area of the parallelogram.

Solutions :

such that a = 6i + 3j + 1k and b = 3i – 1j + 5k

$$a*b\;=\;\text{C}\;=\;\begin{vmatrix}i & j & k \\6 & 3 & 1 \\3 & -1 & -5 \\ \end{vmatrix}$$

$$a*b=i((3*5)–(1*(-1)))+j((1*3)–(6*5))+k((6*(-1))–(3*3))$$

$$a*b=i((15)–(-1)))+j((3)–(30))+k((-6))–(9))$$

$$a*b=i(15+1)+j(3-30)+k(-6-9)$$

$$a*b=16i+j(-27)+k(-15)$$

$$a*b=16i-17j-15k$$