Vector Algebra
Vector Addition and Parallelogram Law
Grade 12
Question:
<p><strong>Ex. 59 (B):</strong> From point P, a man walks 4 units in the North-West direction to reach point Q. Construct parallelogram OPQR with OP and PQ as adjacent sides. Find the position vector of R.</p>
<p>(p) 3</p>
<p>(q) \(\frac{1}{2}(i + 5j)\)</p>
<p>(r) \(2\sqrt{2}(-i + j)\)</p>
<p>(s) \(\frac{2}{3}(i + 5j)\)</p>
Step-by-Step Solution
Key Concept: In a parallelogram OPQR, the fourth vertex R is found using the property that diagonals bisect each other: \(OR = OP + PQ\).
Step 1: North-West direction makes angle 135° with positive x-axis. Step 2: Given \(OR = 4\) and \(\angle XOR = 135°\). Step 3: \(OR = (4\cos 135°)i + (4\sin 135°)j = 4\left(\frac{-i + j}{\sqrt{2}}\right) = 2\sqrt{2}(-i + j)\) ∴ Answer is r: \(2\sqrt{2}(-i + j)\)
Correct Answer: r