The diagonals of a parallelogram are represented by R1 = 3i+2j-7k and R2 = 5i+6j-3k. Find the area of parallelogram.

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

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