Vector Algebra
Position Vectors and Midpoint
Grade 12
Question:
<p>If the position vectors of the points A and B are <strong>i</strong> + 3<strong>j</strong> - <strong>k</strong> and 3<strong>i</strong> - <strong>j</strong> - 3<strong>k</strong>, then what will be the position vector of the mid-point of AB?</p>
<p>(a) <strong>i</strong> + 2<strong>j</strong> - <strong>k</strong></p>
<p>(b) 2<strong>i</strong> + <strong>j</strong> - 2<strong>k</strong></p>
<p>(c) 2<strong>i</strong> + <strong>j</strong> + <strong>k</strong></p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Apply the midpoint formula by averaging corresponding components.
Solution: The mid-point of AB with position vectors A = i + 3 j - k and B = 3 i - j - 3 k is: Mid-point = \(\frac{\mathbf{A} + \mathbf{B}}{2} = \frac{(\mathbf{i} + 3\mathbf{j} - \mathbf{k}) + (3\mathbf{i} - \mathbf{j} - 3\mathbf{k})}{2}\) = \(\frac{4\mathbf{i} + 2\mathbf{j} - 4\mathbf{k}}{2}\) = \(2\mathbf{i} + \mathbf{j} - 2\mathbf{k}\)
Correct Answer: B