Question

If \( \mathbf{a} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \) and \( \mathbf{b} = 2\mathbf{i} - 3\mathbf{j} - 5\mathbf{k} \), then:

• A. \( |\mathbf{a} - \mathbf{b}|>|\mathbf{a}| + |\mathbf{b}| \)
• B. \( |\mathbf{a} - \mathbf{b}|>|\mathbf{b}| - |\mathbf{a}| \)
• C. \( |\mathbf{a} + \mathbf{b}|<|\mathbf{a} - \mathbf{b}| \)
• D. \( |\mathbf{a}| - |\mathbf{b}|>|\mathbf{a} - \mathbf{b}| \)

Hint:

When comparing magnitudes of vectors, calculate each vector's magnitude first, then carefully compare the resulting values step by step.

Answer 1

The Correct Option is B

Given vectors \( \mathbf{a} = \mathbf{i} + 2\mathbf{j} - 3\mathbf{k} \) and \( \mathbf{b} = 2\mathbf{i} - 3\mathbf{j} - 5\mathbf{k} \). Step 1: Compute \( \mathbf{a} - \mathbf{b} \)\[\mathbf{a} - \mathbf{b} = (\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}) - (2\mathbf{i} - 3\mathbf{j} - 5\mathbf{k}) = -\mathbf{i} + 5\mathbf{j} + 2\mathbf{k}.\] Step 2: Determine the magnitude of \( \mathbf{a} - \mathbf{b} \)\[|\mathbf{a} - \mathbf{b}| = \sqrt{(-1)^2 + 5^2 + 2^2} = \sqrt{1 + 25 + 4} = \sqrt{30}.\] Step 3: Calculate the magnitudes of \( \mathbf{a} \) and \( \mathbf{b} \)\[|\mathbf{a}| = \sqrt{1^2 + 2^2 + (-3)^2} = \sqrt{1 + 4 + 9} = \sqrt{14}.\]\[|\mathbf{b}| = \sqrt{2^2 + (-3)^2 + (-5)^2} = \sqrt{4 + 9 + 25} = \sqrt{38}.\] Step 4: Verify the magnitude relationship
Compare \( |\mathbf{a} - \mathbf{b}| \) with \( |\mathbf{b}| - |\mathbf{a}| \):\[|\mathbf{b}| - |\mathbf{a}| = \sqrt{38} - \sqrt{14}.\]Since \( |\mathbf{a} - \mathbf{b}| = \sqrt{30} \) and \( \sqrt{30}>\sqrt{38} - \sqrt{14} \), the inequality \( |\mathbf{a} - \mathbf{b}|>|\mathbf{b}| - |\mathbf{a}| \) is true. Final Answer:\[\boxed{|\mathbf{a} - \mathbf{b}|>|\mathbf{b}| - |\mathbf{a}|}\]