Question:
The area of the rectangle having vertices \(\mathrm{P}, \quad \mathrm{Q}, \quad \mathrm{R}, \quad \mathrm{S}\) with position vectors \(-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}},-\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}\) respectively is
The area of the rectangle having vertices \(\mathrm{P}, \quad \mathrm{Q}, \quad \mathrm{R}, \quad \mathrm{S}\) with position vectors \(-\hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}+\hat{\mathrm{j}}+\hat{\mathrm{k}}, \hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}},-\hat{\mathrm{i}}-\hat{\mathrm{j}}+\hat{\mathrm{k}}\) respectively is
Show Hint
Find the lengths of two adjacent sides from the position vectors and multiply.
Updated On: Nov 13, 2026
- 1 square unit
- 2 square units
- 3 square units
- 4 square units
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
The area of a rectangle is the product of two adjacent side lengths.
Step 2: Side lengths
\(\overrightarrow{PQ} = Q - P = 2\hat{i}\), so \(|PQ| = 2\).
\(\overrightarrow{QR} = R - Q = -2\hat{j}\), so \(|QR| = 2\).
Step 3: Area
Area \(= |PQ|\times|QR| = 2 \times 2 = 4\) square units.
Final Answer:
The area is 4 square units. \[ \boxed{\text{(D) }\text{4 square units}} \]
The area of a rectangle is the product of two adjacent side lengths.
Step 2: Side lengths
\(\overrightarrow{PQ} = Q - P = 2\hat{i}\), so \(|PQ| = 2\).
\(\overrightarrow{QR} = R - Q = -2\hat{j}\), so \(|QR| = 2\).
Step 3: Area
Area \(= |PQ|\times|QR| = 2 \times 2 = 4\) square units.
Final Answer:
The area is 4 square units. \[ \boxed{\text{(D) }\text{4 square units}} \]
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