Vector Algebra
Section Formula and Midpoint
Grade 12
Question:
<p><strong>Ex. 59 (C):</strong> Let M be the mid-point of PQ. Find the position vector of M.</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: Midpoint of two points is the average of their position vectors.
Step 1: Position vector of M (midpoint of PQ) is given by: \(OM = \frac{OP + OQ}{2}\) Step 2: \(OQ = OP + PQ = OP + OR = \frac{3}{\sqrt{2}}(i + j) + 2\sqrt{2}(-i + j)\) Step 3: \(OM = \frac{1}{2}\left[\frac{3}{\sqrt{2}}(i + j) + \frac{3}{\sqrt{2}}(i + j) + 2\sqrt{2}(-i + j)\right] = \frac{1}{2}(i + 5j)\) ∴ Answer is q: \(\frac{1}{2}(i + 5j)\)
Correct Answer: q