Question 49 of 68intermediate🔧 ApplyMCQ1 mark

Area of a rectangle having vertices A, B, C and D with position vectors -î + (1/2)ĵ + 4k̂, î + (1/2)ĵ + 4k̂, î - (1/2)ĵ + 4k̂ and -î - (1/2)ĵ + 4k̂, respectively is

Exercise: EXERCISE 10.4 | Q: 12 | (Page 32)
For More Understanding

Explanation

To find the area of the rectangle, we need to find two adjacent sides and multiply their lengths. The position vectors are given, so we find the vectors representing adjacent sides AB and BC. The area of a rectangle equals the product of the lengths of two adjacent sides.

Solution Steps

  1. Step 1: Find the vector for side AB: AB→=B⃗−A⃗=(i^+12j^+4k^)−(−i^+12j^+4k^)=2i^\overrightarrow{AB} = \vec{B} - \vec{A} = (\hat{i} + \frac{1}{2}\hat{j} + 4\hat{k}) - (-\hat{i} + \frac{1}{2}\hat{j} + 4\hat{k}) = 2\hat{i}

  2. Step 2: Find the vector for side BC: BC→=C⃗−B⃗=(i^−12j^+4k^)−(i^+12j^+4k^)=−j^\overrightarrow{BC} = \vec{C} - \vec{B} = (\hat{i} - \frac{1}{2}\hat{j} + 4\hat{k}) - (\hat{i} + \frac{1}{2}\hat{j} + 4\hat{k}) = -\hat{j}

  3. Step 3: Calculate the magnitudes: ∣AB→∣=∣2i^∣=2|\overrightarrow{AB}| = |2\hat{i}| = 2 and ∣BC→∣=∣−j^∣=1|\overrightarrow{BC}| = |-\hat{j}| = 1

  4. Step 4: Verify perpendicularity: AB→⋅BC→=(2i^)⋅(−j^)=0\overrightarrow{AB} \cdot \overrightarrow{BC} = (2\hat{i}) \cdot (-\hat{j}) = 0, confirming it's a rectangle

  5. Step 5: Calculate area: Area = ∣AB→∣×∣BC→∣=2×1=2|\overrightarrow{AB}| \times |\overrightarrow{BC}| = 2 \times 1 = 2

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