Given diagonals of the parallelogram are:
\(\begin{array}{l}R_{1} = 3\hat{i}+2\hat{j}-7\hat{k} \end{array} \)
,
\(\begin{array}{l}R_{2} = 5\hat{i}+6\hat{j}-3\hat{k}\end{array} \)
As we know, the area of parallelogram is,
\(\begin{array}{l}A = \frac{1}{2}|R_{1}\times R_{2}|\end{array} \)
Finding
\(\begin{array}{l}|R_{1}\times R_{2}|\end{array} \)
:
To find
\(\begin{array}{l}|R_{1}\times R_{2}|\end{array} \)
, we can use matrix method.
Hence,
\(\begin{array}{l}|R_{1}\times R_{2}|\end{array} \)
= \(\begin{array}{l}\begin{vmatrix}\hat{i}&\hat{j} & \hat{k} \\3 & 2 & -7 \\5 & 6& -3 \\\end{vmatrix}\end{array} \)
=
\(\begin{array}{l}= \hat{i}(-6+42)-\hat{j}(-9+35)+\hat{k}(18 -10)\end{array} \)
\(\begin{array}{l}= 36\hat{i}-26\hat{j}+8\hat{k}\end{array} \)
Now, find the magnitude of
\(\begin{array}{l}|R_{1}\times R_{2}|\end{array} \)
.
\(\begin{array}{l}= \sqrt{(36)^{2}+(-26)^{2}+(8)^{2}}\end{array} \)
=
\(\begin{array}{l}\sqrt{2036}\end{array} \)
Magnitude of
\(\begin{array}{l}|R_{1}\times R_{2}|\end{array} \)
= 45.12
Hence,
\(\begin{array}{l}A = \frac{1}{2}|R_{1}\times R_{2}|\end{array} \)
= (½)(45.12)
A = 22.56
Therefore, the area of a parallelogram is 22.56 square units
