Question

Show that area of the parallelogram whose diagonals are given by \(\vec a \, and \, \vec b \) is \(\frac{|\vec a \times \vec b|}{2}\) Also find the area of the parallelogram whose diagonals are \(2 \hat i - \hat j + \hat k \,and\, \hat i + 3 \hat j - \hat k.\)

Answer 1

We have,

Let ABCD be a parallelogram.

In ABCD,

Now, by triangle law of addition, we get

Adding equations (i) and (ii), we get

We know that,

Vector area of parallelogram ABCD is given by,

Now, we need to find the area of parallelogram whose diagonals are \(2 \hat i - \hat j + \hat k \,and\, \hat i + 3 \hat j - \hat k.\)

We have already derived the relationship between area of parallelogram and diagonals of parallelogram, which is