<p>If the position vectors of the vertices of a triangle be \(2\hat{i} + 4\hat{j} - \hat{k}\), \(4\hat{i} + 5\hat{j} + \hat{k}\) and \(3\hat{i} + 6\hat{j} - 3\hat{k}\), then the triangle is</p>
Step-by-Step Solution
Key Concept: Find the side vectors using position vectors and calculate their magnitudes to classify the triangle type
Solution: Let A, B, C be the vertices with position vectors: \(\overrightarrow{OA} = 2\hat{i} + 4\hat{j} - \hat{k}\), \(\overrightarrow{OB} = 4\hat{i} + 5\hat{j} + \hat{k}\), \(\overrightarrow{OC} = 3\hat{i} + 6\hat{j} - 3\hat{k}\) \(\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA} = 2\hat{i} + \hat{j} + 2\hat{k}\) Calculate \(|\overrightarrow{AB}|\), \(|\overrightarrow{BC}|\), \(|\overrightarrow{CA}|\) and compare. If all three sides are equal, the triangle is equilateral.
Correct Answer: C