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Step-by-Step Solution
Step 1: Identify the Given Vectors
We are given the diagonals of the parallelogram as two vectors:
$\overrightarrow{d_1} = 8\hat{i} - 6\hat{j} + 0\hat{k}$
$\overrightarrow{d_2} = 3\hat{i} + 4\hat{j} - 12\hat{k}$
Step 2: Recall the Formula for the Area of the Parallelogram
The area of the parallelogram with diagonals $\overrightarrow{d_1}$ and $\overrightarrow{d_2}$ is given by:
$ \text{Area} \;=\; \frac{1}{2} \bigl\lvert \,\overrightarrow{d_1} \times \overrightarrow{d_2}\,\bigr\rvert
$
Step 3: Compute the Cross Product $\overrightarrow{d_1} \times \overrightarrow{d_2}$
Use the determinant form to find the cross product:
$
\overrightarrow{d_1} \times \overrightarrow{d_2}
=
\begin{vmatrix}
\hat{i} & \hat{j} & \hat{k} \\
8 & -6 & 0 \\
3 & 4 & -12
\end{vmatrix}
$
Expanding this determinant:
Coefficient of $\hat{i}$: $(-6)(-12) - (0)(4) = 72$
Coefficient of $\hat{j}$: $-\bigl(8(-12) - 0 \cdot 3 \bigr)
= -(-96) = 96$
Coefficient of $\hat{k}$: $(8)(4) - (-6)(3) = 32 + 18 = 50$
Hence,
$
\overrightarrow{d_1} \times \overrightarrow{d_2}
= 72\,\hat{i} + 96\,\hat{j} + 50\,\hat{k}.
$
Step 4: Find the Magnitude of the Cross Product
$
\bigl\lvert \,\overrightarrow{d_1} \times \overrightarrow{d_2}\,\bigr\rvert
= \sqrt{(72)^2 + (96)^2 + (50)^2}
= \sqrt{5184 + 9216 + 2500}
= \sqrt{16900}
= 130.
$
Step 5: Calculate the Area of the Parallelogram
Finally, the area of the parallelogram is given by half of this magnitude:
$
\text{Area}
= \frac{1}{2} \times 130
= 65.
$
Therefore, the area of the parallelogram is 65 square units.
