Vector Algebra
Section Formula and Collinearity
Grade 12
Question:
<p><strong>Ex. 59 (D):</strong> If the line OM meets the diagonal PR in the point T, then find OT.</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: When a line from O meets diagonal PR at T, use the section formula with the ratio determined by collinearity conditions.
Step 1: Point T lies on PR such that PT : RT = 1 : 2. Step 2: Using section formula: \(OT = \frac{1 \cdot OR + 2 \cdot OP}{3}\) Step 3: \(OT = \frac{1}{3}\left[2\sqrt{2}(-i + j) + 2 \cdot \frac{3}{\sqrt{2}}(i + j)\right]\) Step 4: \(OT = \frac{1}{3}\left[2\sqrt{2}(-i + j) + \frac{6}{\sqrt{2}}(i + j)\right] = \frac{2}{3}(i + 5j)\) ∴ Answer is s: \(\frac{2}{3}(i + 5j)\)
Correct Answer: s