Vector Algebra
Position Vector and Direction Cosines
Grade 12
Question:
<p><strong>Ex. 59 (A):</strong> In the Cartesian plane, a man starts at origin and walks a distance of 3 units in the North-East direction and reaches a point P. Find the position vector of P.</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: Use polar form of position vector: magnitude times unit direction vector. North-East is 45° from horizontal.
Step 1: Let \(i\) and \(j\) be unit vectors along OX and OY respectively. Step 2: Given \(OP = 3\) and \(\angle XOP = 45°\). Step 3: \(OP = (3\cos 45°)i + (3\sin 45°)j = 3\left(\frac{i + j}{\sqrt{2}}\right) = \frac{3}{\sqrt{2}}(i + j)\) ∴ Answer is p: \(\frac{3}{\sqrt{2}}(i + j)\)
Correct Answer: p