Vector Algebra
Magnitude of Vectors
Grade None
Question:
<p>If <span class="math">|\mathbf{a}| = 5</span>, <span class="math">|\mathbf{a} - \mathbf{b}| = 8</span> and <span class="math">|\mathbf{a} + \mathbf{b}| = 10</span>, then <span class="math">|\mathbf{b}|</span> is equal to</p>
<p>(a) 1</p>
<p>(b) <span class="math">\sqrt{57}</span></p>
<p>(c) 3</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use the parallelogram law identity to relate the magnitudes of sum and difference of vectors to find the unknown magnitude.
Solution: We know that for any two vectors: |\mathbf{a} + \mathbf{b}|^2 + |\mathbf{a} - \mathbf{b}|^2 = 2(|\mathbf{a}|^2 + |\mathbf{b}|^2) Substituting the given values: (10)^2 + (8)^2 = 2[(5)^2 + |\mathbf{b}|^2] 100 + 64 = 50 + 2|\mathbf{b}|^2 164 = 50 + 2|\mathbf{b}|^2 |\mathbf{b}|^2 = 57 \therefore |\mathbf{b}| = \sqrt{57}
Correct Answer: B