Key Concepts and Formulas

• Area of a Parallelogram using Diagonals: If $\overrightarrow{d_1}$ and $\overrightarrow{d_2}$ are the vectors representing the diagonals of a parallelogram, then the area (A) of the parallelogram is given by: $A = \frac{1}{2} |\overrightarrow{d_1} \times \overrightarrow{d_2}|$ This formula arises because the magnitude of the cross product of two vectors is equal to the area of the parallelogram formed by those vectors as adjacent sides. When the given vectors are diagonals, the area of the parallelogram they form is half the area of the parallelogram formed by using them as adjacent sides.
• Cross Product of Two Vectors: For vectors $\vec{A} = A_x\widehat i + A_y\widehat j + A_z\widehat k$ and $\vec{B} = B_x\widehat i + B_y\widehat j + B_z\widehat k$, their cross product is calculated as: $\vec{A} \times \vec{B} = \begin{vmatrix} \widehat i & \widehat j & \widehat k \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$
• Magnitude of a Vector: The magnitude of a vector $\vec{V} = V_x\widehat i + V_y\widehat j + V_z\widehat k$ is given by: $|\vec{V}| = \sqrt{V_x^2 + V_y^2 + V_z^2}$

Step-by-Step Solution

We are given the vectors representing the diagonals of a parallelogram: $\overrightarrow{d_1} = 8\widehat i - 6\widehat j$ $\overrightarrow{d_2} = 3\widehat i + 4\widehat j - 12\widehat k$

Step 1: Express the first diagonal vector in 3D form. To perform the cross product, it's convenient to have both vectors in 3D. The first vector, $\overrightarrow{d_1}$, can be written as: $\overrightarrow{d_1} = 8\widehat i - 6\widehat j + 0\widehat k$

Step 2: Calculate the cross product of the diagonal vectors ($\overrightarrow{d_1} \times \overrightarrow{d_2}$). Using the determinant formula for the cross product: $\overrightarrow{d_1} \times \overrightarrow{d_2} = \begin{vmatrix} \widehat i & \widehat j & \widehat k \\ 8 & -6 & 0 \\ 3 & 4 & -12 \end{vmatrix}$ Expanding the determinant:

• The $\widehat i$ component is: $((-6) \times (-12) - (0) \times 4)\widehat i = (72 - 0)\widehat i = 72\widehat i$
• The $\widehat j$ component is: $-((8) \times (-12) - (0) \times 3)\widehat j = -(-96 - 0)\widehat j = -(-96)\widehat j = 96\widehat j$
• The $\widehat k$ component is: $((8) \times 4 - (-6) \times 3)\widehat k = (32 - (-18))\widehat k = (32 + 18)\widehat k = 50\widehat k$ Therefore, the cross product vector is: $\overrightarrow{d_1} \times \overrightarrow{d_2} = 72\widehat i + 96\widehat j + 50\widehat k$

Step 3: Calculate the magnitude of the cross product vector. The magnitude of the vector $\overrightarrow{d_1} \times \overrightarrow{d_2}$ is: $|\overrightarrow{d_1} \times \overrightarrow{d_2}| = \sqrt{(72)^2 + (96)^2 + (50)^2}$ Let's compute the squares:

• $72^2 = 5184$
• $96^2 = 9216$
• $50^2 = 2500$ Now, sum these values: $|\overrightarrow{d_1} \times \overrightarrow{d_2}| = \sqrt{5184 + 9216 + 2500} = \sqrt{16900}$ To find the square root of 16900: $\sqrt{16900} = \sqrt{169 \times 100} = \sqrt{169} \times \sqrt{100} = 13 \times 10 = 130$ So, the magnitude of the cross product is 130.

Step 4: Calculate the area of the parallelogram. Using the formula for the area of a parallelogram with diagonal vectors: $A = \frac{1}{2} |\overrightarrow{d_1} \times \overrightarrow{d_2}|$ Substitute the magnitude calculated in Step 3: $A = \frac{1}{2} \times 130$ $A = 65$ The area of the parallelogram is 65 square units.

Common Mistakes & Tips

• Forgetting the $\frac{1}{2}$ Factor: When the given vectors are diagonals, always remember to multiply the magnitude of their cross product by $\frac{1}{2}$. If the vectors were adjacent sides, the area would be $|\overrightarrow{d_1} \times \overrightarrow{d_2}|$.
• Sign Errors in Cross Product: Be meticulous with the signs when expanding the determinant for the cross product, especially the $-\widehat j$ term. A single sign error can lead to a completely incorrect result.
• Arithmetic Errors: Squaring numbers and summing them can be prone to arithmetic mistakes. Double-checking these calculations is important.

Summary

The area of a parallelogram can be calculated using its diagonal vectors by taking half the magnitude of their cross product. We first calculated the cross product of the given diagonal vectors, $\overrightarrow{d_1} = 8\widehat i - 6\widehat j$ and $\overrightarrow{d_2} = 3\widehat i + 4\widehat j - 12\widehat k$, which resulted in $72\widehat i + 96\widehat j + 50\widehat k$. Then, we found the magnitude of this cross product to be 130. Finally, applying the formula $A = \frac{1}{2} |\overrightarrow{d_1} \times \overrightarrow{d_2}|$, we determined the area of the parallelogram to be 65 square units. This corresponds to option (B).

The final answer is $\boxed{65}$.