Vector Algebra
Vector properties and geometry
Grade 12
Question:
<p><strong>Ex. 46 Statement I:</strong> If \(|\mathbf{a} + \mathbf{b}| = |\mathbf{a} - \mathbf{b}|\), then \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular to each other.</p><p><strong>Statement II:</strong> If the diagonals of a parallelogram are equal in magnitude, then the parallelogram is a rectangle.</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: In a parallelogram with sides as vectors, equal diagonals imply perpendicular sides, making it a rectangle.
\(\mathbf{a} + \mathbf{b}\) and \(\mathbf{a} - \mathbf{b}\) are the diagonals of a parallelogram whose sides are \(\mathbf{a}\) and \(\mathbf{b}\). If the diagonals are equal in magnitude, the parallelogram is a rectangle, which means \(\mathbf{a} \perp \mathbf{b}\). ∴ Answer is (a).
Correct Answer: a