Vector Algebra
Vector magnitude and properties
Grade 12
Question:
<p><strong>Ex. 45 Statement I:</strong> If \(|\mathbf{a}| = 3\), \(|\mathbf{b}| = 4\) and \(|\mathbf{a} + \mathbf{b}| = 5\), then \(|\mathbf{a} - \mathbf{b}| = 5\).</p><p><strong>Statement II:</strong> The length of the diagonals of a rectangle is the same.</p>
<p>(a) Both Statement I and Statement II are correct and Statement II is the correct explanation of Statement I</p>
<p>(b) Both Statement I and Statement II are correct but Statement II is not the correct explanation of Statement I</p>
<p>(c) Statement I is correct but Statement II is incorrect</p>
<p>(d) Statement II is correct but Statement I is incorrect</p>
Step-by-Step Solution
Key Concept: A parallelogram with perpendicular sides is a rectangle, and a rectangle has equal diagonals.
We have adjacent sides of triangle \(|\mathbf{a}| = 3\), \(|\mathbf{b}| = 4\). The length of the diagonal is \(|\mathbf{a} + \mathbf{b}| = 5\). Since it satisfies the Pythagoras theorem, \(\mathbf{a} \perp \mathbf{b}\). So, the parallelogram is a rectangle. Hence, the length of the other diagonal is \(|\mathbf{a} - \mathbf{b}| = 5\). Both statements are correct and Statement II explains Statement I. ∴ Answer is (a).
Correct Answer: a