Vector Algebra
Position Vectors and Division
Grade 12
Question:
<p>If <strong>a</strong> and <strong>b</strong> are the position vectors of A and B respectively, then the position vector of a point C on AB produced such that AC = 3 AB is</p>
<p>(a) 3<strong>a</strong> - <strong>b</strong></p>
<p>(b) 3<strong>b</strong> - <strong>a</strong></p>
<p>(c) 3<strong>a</strong> - 2<strong>b</strong></p>
<p>(d) 3<strong>b</strong> - 2<strong>a</strong></p>
Step-by-Step Solution
Key Concept: External division formula: if a point divides a line segment externally in ratio m:n, its position vector is (m·b - n·a)/(m - n).
Step 1: Since AC = 3 AB, point C divides AB externally in the ratio AC : BC = 3 : 2. Step 2: For external division, if C divides AB externally in ratio 3 : 2, then the position vector of C is: \(\mathbf{c} = \frac{3\mathbf{b} - 2\mathbf{a}}{3 - 2} = 3\mathbf{b} - 2\mathbf{a}\) ∴ Answer is (d).
Correct Answer: D