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 two 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: Calculate \( \mathbf{a} - \mathbf{b} \) \[ \mathbf{a} - \mathbf{b} = (\mathbf{i} + 2\mathbf{j} - 3\mathbf{k}) - (2\mathbf{i} - 3\mathbf{j} - 5\mathbf{k}). \] Simplifying this expression: \[ \mathbf{a} - \mathbf{b} = -\mathbf{i} + 5\mathbf{j} + 2\mathbf{k}. \] Step 2: Find the magnitude of \( \mathbf{a} - \mathbf{b} \) The magnitude is given by: \[ |\mathbf{a} - \mathbf{b}| = \sqrt{(-1)^2 + 5^2 + 2^2}. \] Simplifying further: \[ |\mathbf{a} - \mathbf{b}| = \sqrt{1 + 25 + 4} = \sqrt{30}. \] Step 3: Calculate \( |\mathbf{a}| \) and \( |\mathbf{b}| \) For \( |\mathbf{a}| \), we have: \[ |\mathbf{a}| = \sqrt{1^2 + 2^2 + (-3)^2} = \sqrt{1 + 4 + 9} = \sqrt{14}. \] For \( |\mathbf{b}| \), we compute: \[ |\mathbf{b}| = \sqrt{2^2 + (-3)^2 + (-5)^2} = \sqrt{4 + 9 + 25} = \sqrt{38}. \] Step 4: Verify the relationship between the magnitudes
Now, we compare \( |\mathbf{a} - \mathbf{b}| \) with \( |\mathbf{b}| - |\mathbf{a}| \): \[ |\mathbf{b}| - |\mathbf{a}| = \sqrt{38} - \sqrt{14}. \] Since we know that \( |\mathbf{a} - \mathbf{b}| = \sqrt{30} \), and \( \sqrt{30}>\sqrt{38} - \sqrt{14} \), the inequality holds true: \[ |\mathbf{a} - \mathbf{b}|>|\mathbf{b}| - |\mathbf{a}|. \] Final Answer: \[ \boxed{|\mathbf{a} - \mathbf{b}|>|\mathbf{b}| - |\mathbf{a}|} \]