Vector Algebra
Position Vectors
Grade 12
Question:
<p>The position vector of a point C with respect to B is \(\mathbf{i} + \mathbf{j}\) and that of B with respect to A is \(\mathbf{i} - \mathbf{j}\). Find the position vector of C with respect to A.</p>
<p>(a) \(2\mathbf{i}\)</p>
<p>(b) \(2\mathbf{j}\)</p>
<p>(c) \(-2\mathbf{j}\)</p>
<p>(d) \(-2\mathbf{i}\)</p>
Step-by-Step Solution
Key Concept: Position vectors are additive: the vector from A to C equals the sum of vectors from A to B and B to C.
Step 1: Position vector of C with respect to B: \(\overrightarrow{BC} = \mathbf{i} + \mathbf{j}\) Step 2: Position vector of B with respect to A: \(\overrightarrow{AB} = \mathbf{i} - \mathbf{j}\) Step 3: Using the vector addition property: \(\overrightarrow{AC} = \overrightarrow{AB} + \overrightarrow{BC}\) Step 4: \(\overrightarrow{AC} = (\mathbf{i} - \mathbf{j}) + (\mathbf{i} + \mathbf{j})\) Step 5: \(\overrightarrow{AC} = 2\mathbf{i}\) ∴ Answer is (a) \(2\mathbf{i}\)
Correct Answer: a