Question

If vector \( \mathbf{a} = 3 \hat{i} + 2 \hat{j} - \hat{k} \) \text{ and } \( \mathbf{b} = \hat{i} - \hat{j} + \hat{k} \),  then which of the following is correct?

• A. \( \mathbf{a} \parallel \mathbf{b} \)
• B. \( \mathbf{a} \perp \mathbf{b} \)
• C. \( |\mathbf{a}|>|\mathbf{b}| \)
• D. \( |\mathbf{a}| = |\mathbf{b}| \)

Hint:

The magnitude of a vector is calculated by taking the square root of the sum of the squares of its components. For vectors, use the formula \( |\mathbf{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2} \).

Answer 1

The Correct Option is C

First, let's calculate the magnitudes of the vectors \( \mathbf{a} \) and \( \mathbf{b} \). The magnitude of a vector \( \mathbf{v} = v_x \hat{i} + v_y \hat{j} + v_z \hat{k} \) is given by: \[ |\mathbf{v}| = \sqrt{v_x^2 + v_y^2 + v_z^2} \] Magnitude of vector \( \mathbf{a} \): \[ \mathbf{a} = 3 \hat{i} + 2 \hat{j} - \hat{k} \] The magnitude of \( \mathbf{a} \) is: \[ |\mathbf{a}| = \sqrt{3^2 + 2^2 + (-1)^2} = \sqrt{9 + 4 + 1} = \sqrt{14} \] Magnitude of vector \( \mathbf{b} \): \[ \mathbf{b} = \hat{i} - \hat{j} + \hat{k} \] The magnitude of \( \mathbf{b} \) is: \[ |\mathbf{b}| = \sqrt{1^2 + (-1)^2 + 1^2} = \sqrt{1 + 1 + 1} = \sqrt{3} \] Comparison of magnitudes: \[ |\mathbf{a}| = \sqrt{14} \quad \text{and} \quad |\mathbf{b}| = \sqrt{3} \] Since \( \sqrt{14}>\sqrt{3} \), we can conclude that \( |\mathbf{a}|>|\mathbf{b}| \). Therefore, the correct answer is (C).