Question

Let $ABCD$ be a rectangle. If $\overrightarrow{AB} = 5\hat{i} + 4\hat{j} - 3\hat{k}$ and $\overrightarrow{AD} = 3\hat{i} + 2\hat{j} - \hat{k}$, then the length of $BD$ is

• A. $2\sqrt{5}$
• B. $3\sqrt{2}$
• C. $4\sqrt{3}$
• D. $2\sqrt{3}$
• E. $3\sqrt{3}$

Hint:

In a rectangle $ABCD$, the diagonal $\overrightarrow{BD} = \overrightarrow{BA} + \overrightarrow{AD}$. Don't confuse $\overrightarrow{BD}$ with $\overrightarrow{BC} + \overrightarrow{CD}$; both routes give the same result by vector addition.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
In a rectangle, $\overrightarrow{BD} = \overrightarrow{BA} + \overrightarrow{AD} = -\overrightarrow{AB} + \overrightarrow{AD}$. Compute the magnitude of this diagonal vector.

Step 2: Detailed Explanation:
\[ \overrightarrow{BD} = -\overrightarrow{AB} + \overrightarrow{AD} = -(5\hat{i}+4\hat{j}-3\hat{k}) + (3\hat{i}+2\hat{j}-\hat{k}) \] \[ = (-5+3)\hat{i} + (-4+2)\hat{j} + (3-1)\hat{k} = -2\hat{i} - 2\hat{j} + 2\hat{k} \] \[ |BD| = \sqrt{(-2)^2+(-2)^2+(2)^2} = \sqrt{4+4+4} = \sqrt{12} = 2\sqrt{3} \]

Step 3: Final Answer:
Length of $BD = 2\sqrt{3}$.