Question

Define cross product of two vectors. Show that the magnitude of cross product of two vectors is numerically equal to the area of a parallelogram whose adjacent sides represent the two vectors.

Answer 1

Answer with Explanation:

Let A and B are two vectors

$A=a_1i+a_2j+a_3k$

$B=b_1i+b_2j+b_3k$

Cross product of two vectors A and B is given by

$A\times B=\begin{vmatrix}i&j&k\\a_1&a_2&a_3\\b_1&b_2&b_3\end{vmatrix}$

Magnitude of cross product=$\mid A\times B\mid$

If adjacent sides of parallelogram are A and B

Then, area of parallelogram is given by

$\mid A\times B\mid$

Hence, the magnitude of cross product of two vectors is numerically equal to the area of parallelogram.

Hence, proved.