Vector Algebra
Position Vectors and Division
Grade 12
Question:
<p>If position vector of point A is <strong>a</strong> + 2<strong>b</strong> and <strong>a</strong> divides AB in the ratio 2 : 3, then the position vector of B is</p>
<p>(a) 2<strong>a</strong> - <strong>b</strong></p>
<p>(b) <strong>b</strong> - 2<strong>a</strong></p>
<p>(c) <strong>a</strong> - 3<strong>b</strong></p>
<p>(d) <strong>b</strong></p>
Step-by-Step Solution
Key Concept: Use the section formula for internal division: if a point P divides AB in ratio m:n, then position vector of P = (n·A + m·B)/(m + n).
Step 1: If x is the position vector of B, and a divides AB in the ratio 2 : 3, then by the section formula: \(\mathbf{a} + 2\mathbf{b} = \frac{2\mathbf{x} + 3(\mathbf{a} + 2\mathbf{b})}{2 + 3}\) Step 2: \(5(\mathbf{a} + 2\mathbf{b}) = 2\mathbf{x} + 3\mathbf{a} + 6\mathbf{b}\) Step 3: \(5\mathbf{a} + 10\mathbf{b} = 2\mathbf{x} + 3\mathbf{a} + 6\mathbf{b}\) Step 4: \(2\mathbf{x} = 2\mathbf{a} + 4\mathbf{b}\) \(\Rightarrow\) \(\mathbf{x} = \mathbf{a} - 3\mathbf{b}\) ∴ Answer is (c).
Correct Answer: C