Question

The vectors \[ \vec{a} = 2\hat{i} - \hat{j} + \hat{k}, \quad \vec{b} = \hat{i} - 3\hat{j} - 5\hat{k}, \quad \vec{c} = -3\hat{i} + 4\hat{j} + 4\hat{k} \] 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

The magnitude of a vector \( \mathbf{v} = x\hat{i} + y\hat{j} + z\hat{k} \) is given by:
\[ |\mathbf{v}| = \sqrt{x^2 + y^2 + z^2} \]
1. Magnitude of \( \mathbf{a} \):
\[ |\mathbf{a}| = \sqrt{(2)^2 + (-1)^2 + (1)^2} = \sqrt{6} \]
2. Magnitude of \( \mathbf{b} \):
\[ |\mathbf{b}| = \sqrt{(1)^2 + (-3)^2 + (-5)^2} = \sqrt{35} \]
3. Magnitude of \( \mathbf{c} \):
\[ |\mathbf{c}| = \sqrt{(-3)^2 + (4)^2 + (4)^2} = \sqrt{41} \]
A triangle is a right-angled triangle if the dot product between two vectors representing two sides of the triangle is zero. Let's calculate the dot product of the vectors \( \mathbf{a} \) and \( \mathbf{b} \), \( \mathbf{b} \) and \( \mathbf{c} \), and \( \mathbf{c} \) and \( \mathbf{a} \).

1. Dot product \( \mathbf{a} \cdot \mathbf{b} \):
\[ \mathbf{a} \cdot \mathbf{b} = (2)(1) + (-1)(-3) + (1)(-5) = 0 \]
Since \( \mathbf{a} \cdot \mathbf{b} = 0 \), the vectors \( \mathbf{a} \) and \( \mathbf{b} \) are perpendicular, meaning the angle between them is \( 90^\circ \).