Let A(2,3,5) and C(-3,4,-2) be opposite vertices of a parallelogram ABCD.

Question

Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram $ABCD$. If the diagonal $\vec{BD} = \hat i+2\hat j+3\hat k$, then the area of the parallelogram is equal to

Accepted Answer

The area of a parallelogram in terms of its diagonals $d_1=\vec{AC}$, $d_2=\vec{BD}$ is $\dfrac12|d_1\times d_2|$.

$\vec{AC} = C-A = (-5,1,-7)$, $\vec{BD}=(1,2,3)$.

$$\vec{AC}\times\vec{BD} = \begin{vmatrix}\hat i&\hat j&\hat k\-5&1&-7\1&2&3\end{vmatrix} = (3+14)\hat i-(-15+7)\hat j+(-10-1)\hat k = (17,8,-11)$$

$$|\vec{AC}\times\vec{BD}| = \sqrt{289+64+121}=\sqrt{474}$$

$$\text{Area} = \frac12\sqrt{474}$$

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