Vector Algebra
Magnitude of Vectors
Grade 12
Question:
<p>If \(|\vec{a}| = 2\), \(|\vec{b}| = 3\) and \(|2\vec{a} - \vec{b}| = 5\), then \(|2\vec{a} + \vec{b}|\) equals</p>
<p>17</p>
<p>7</p>
<p>5</p>
<p>1</p>
Step-by-Step Solution
Key Concept: Use the identity |u + v|² = |u|² + |v|² + 2u·v to find the dot product from the given condition, then apply it again to find the required magnitude.
Step 1: Expand |2a - b|^2 using the formula |u|^2 = u·u |2a - b|^2 = (2a - b)·(2a - b) = 4|a|^2 - 4a·b + |b|^2 = 25 Step 2: Substitute |a| = 2 and |b| = 3 4(4) - 4a·b + 9 = 25 16 - 4a·b + 9 = 25 -4a·b = 0 ∴ a·b = 0 Step 3: Find |2a + b|^2 using the same approach |2a + b|^2 = (2a + b)·(2a + b) = 4|a|^2 + 4a·b + |b|^2 |2a + b|^2 = 4(4) + 4(0) + 9 = 16 + 9 = 25 Step 4: Take the square root |2a + b| = √25 = 5 ∴ Answer: C
Correct Answer: C