Vector Algebra
Scalar Product of Vectors
Grade 12
Question:
<p><span class="math">\((\vec{a} \cdot \vec{i})\vec{i} + (\vec{a} \cdot \vec{j})\vec{j} + (\vec{a} \cdot \vec{k})\vec{k}\)</span> is equal to</p>
<p>(a) <span class="math">\(\vec{a}\)</span></p>
<p>(b) <span class="math">\(2\vec{a}\)</span></p>
<p>(c) <span class="math">\(3\vec{a}\)</span></p>
<p>(d) <span class="math">\(0\)</span></p>
Step-by-Step Solution
Key Concept: The scalar product of a vector with unit vectors along coordinate axes gives its components. Reconstructing the vector from its components yields the original vector.
Solution: Let \(\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}\) \(\vec{a} \cdot \hat{i} = (a_1\hat{i} + a_2\hat{j} + a_3\hat{k}) \cdot \hat{i} = a_1\) (since \(\hat{i} \cdot \hat{i} = 1\) , \(\hat{j} \cdot \hat{i} = 0\) , \(\hat{k} \cdot \hat{i} = 0\) ) \(\vec{a} \cdot \hat{j} = a_2\) \(\vec{a} \cdot \hat{k} = a_3\) Therefore: \((\vec{a} \cdot \hat{i})\hat{i} + (\vec{a} \cdot \hat{j})\hat{j} + (\vec{a} \cdot \hat{k})\hat{k} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} = \vec{a}\) ∴ Answer is (a).
Correct Answer: A