Question

Adjacent sides of a parallelogram are given by \( 6\hat{i} - \hat{j} + 5\hat{k} \) and \( \hat{i} + 5\hat{j} - 2\hat{k} \). Find the area of parallelogram.

Hint:

Be careful not to divide by 2! Dividing by 2 gives the area of a {triangle} formed by the same side vectors. Area of Parallelogram = \( |\vec{a} \times \vec{b}| \).

Answer 1

Step 1: Understanding the Concept:
The area of a parallelogram formed by adjacent side vectors \( \vec{a} \) and \( \vec{b} \) is the magnitude of their cross product, \( |\vec{a} \times \vec{b}| \).
Step 2: Detailed Explanation:
Let \( \vec{a} = 6\hat{i} - \hat{j} + 5\hat{k} \) and \( \vec{b} = \hat{i} + 5\hat{j} - 2\hat{k} \).
Calculate \( \vec{a} \times \vec{b} \):
\[ \vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 6 & -1 & 5 \\ 1 & 5 & -2 \end{vmatrix} \] \[ = \hat{i}(2 - 25) - \hat{j}(-12 - 5) + \hat{k}(30 - (-1)) \] \[ = -23\hat{i} + 17\hat{j} + 31\hat{k} \] Now calculate the magnitude:
\[ \text{Area} = \sqrt{(-23)^2 + 17^2 + 31^2} = \sqrt{529 + 289 + 961} = \sqrt{1779} \] Step 3: Final Answer:
The area is \( \sqrt{1779} \) sq. units.