Vector Algebra
Section Formula and Division of Line Segment
Grade 12
Question:
<p>The position vector of the points which divides internally in the ratio 2 : 3 the join of the points 2<strong>a</strong> - 3<strong>b</strong> and 3<strong>a</strong> - 2<strong>b</strong>, is</p>
<p>(a) \(\frac{12}{5}\mathbf{a} + \frac{13}{5}\mathbf{b}\)</p>
<p>(b) \(\frac{12}{5}\mathbf{a} - \frac{13}{5}\mathbf{b}\)</p>
<p>(c) \(\frac{3}{5}\mathbf{a} - \frac{2}{5}\mathbf{b}\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Apply the section formula for internal division with vectors.
Solution: Using the section formula, if a point divides the line segment joining points with position vectors p and q in the ratio m:n internally, then the position vector is \(\frac{m\mathbf{q} + n\mathbf{p}}{m+n}\) Here, p = 2 a - 3 b , q = 3 a - 2 b , m = 2, n = 3 Position vector = \(\frac{2(3\mathbf{a} - 2\mathbf{b}) + 3(2\mathbf{a} - 3\mathbf{b})}{2+3}\) = \(\frac{6\mathbf{a} - 4\mathbf{b} + 6\mathbf{a} - 9\mathbf{b}}{5}\) = \(\frac{12\mathbf{a} - 13\mathbf{b}}{5}\) = \(\frac{12}{5}\mathbf{a} - \frac{13}{5}\mathbf{b}\)
Correct Answer: C