Vector Algebra
Angle Between Vectors
Grade 12
Question:
<p>A vector equally inclined to the vectors \(\mathbf{i} - \mathbf{j} + \mathbf{k}\) and \(\mathbf{i} + \mathbf{j} - \mathbf{k}\) then the plane containing them is</p>
<p>(a) \(\frac{\mathbf{i} + \mathbf{j} - \mathbf{k}}{3}\)</p>
<p>(b) \(\mathbf{j} - \mathbf{k}\)</p>
<p>(c) \(2\mathbf{i}\)</p>
<p>(d) \(\mathbf{i}\)</p>
Step-by-Step Solution
Key Concept: A vector equally inclined to two vectors lies along their sum or bisects the angle between them.
Step 1: A vector equally inclined to \(\mathbf{a} = \mathbf{i} - \mathbf{j} + \mathbf{k}\) and \(\mathbf{b} = \mathbf{i} + \mathbf{j} - \mathbf{k}\) must be proportional to \(\mathbf{a} + \mathbf{b} = 2\mathbf{i}\) or \(\mathbf{a} - \mathbf{b} = -2\mathbf{j} + 2\mathbf{k}\). Step 2: The vector \(\mathbf{i}\) is proportional to \(2\mathbf{i}\) and makes equal angles with both vectors.
Correct Answer: D