Question

The vectors \( \vec{a} = 2\hat{i} - \hat{j} + \hat{k} \), \( \vec{b} = \hat{i} - 3\hat{j} - 5\hat{k} \), and \( \vec{c} = -3\hat{i} + 4\hat{j} + 4\hat{k} \) represent the sides of:

• A. an equilateral triangle
• B. an obtuse-angled triangle
• C. an isosceles triangle
• D. a right-angled triangle

Hint:

Use dot products to identify right angles in vector triangles.

Answer 1

The Correct Option is D

Step 1: Determine vector magnitudes.
Calculate the magnitudes: \[ |\vec{a}| = \sqrt{2^2 + (-1)^2 + 1^2} = \sqrt{6}, \,\] \[|\vec{b}| = \sqrt{1^2 + (-3)^2 + (-5)^2} = \sqrt{35}, \,\]\[ |\vec{c}| = \sqrt{(-3)^2 + 4^2 + 4^2} = \sqrt{41}. \]
Step 2: Verify for a right angle via dot products.
Compute \( \vec{a} \cdot \vec{b} \), \( \vec{b} \cdot \vec{c} \), and \( \vec{c} \cdot \vec{a} \). A zero result indicates a right-angled triangle. For instance: \[ \vec{a} \cdot \vec{c} = (2)(-3) + (-1)(4) + (1)(4) = -6 - 4 + 4 = 0. \] 
Step 3: Classify the triangle.
As \( \vec{a} \cdot \vec{c} = 0 \), the triangle is right-angled.