Vector Algebra
Dot product
Grade 12
Question:
<p>We have <span>\(|2\vec{a} - \vec{b}|^2 = 25 \Rightarrow 4a^2 + b^2 - 4\vec{a}\cdot\vec{b} = 25\)</span>. Given <span>\(|\vec{a}| = 2, |\vec{b}| = 3\)</span>, find <span>\(|2\vec{a} + \vec{b}|^2\)</span>:</p>
<p>\(25\)</p>
<p>\(49\)</p>
<p>\(169\)</p>
<p>\(0\)</p>
Step-by-Step Solution
Key Concept: Expand dot products using the formula |u + v|² = |u|² + |v|² + 2(u·v), then use the constraint equation to find the dot product a·b, which transfers to the target expression.
Step 1: Expand the constraint |2 a - b |^2 = 25 |2 a - b |^2 = (2 a - b ) · (2 a - b ) = 4| a |^2 - 4 a · b + | b |^2 = 25 Step 2: Substitute | a | = 2 and | b | = 3 4(4) - 4 a · b + 9 = 25 16 - 4 a · b + 9 = 25 25 - 4 a · b = 25 a · b = 0 Step 3: Find |2 a + b |^2 |2 a + b |^2 = 4| a |^2 + 4 a · b + | b |^2 = 4(4) + 4(0) + 9 = 16 + 0 + 9 = 25 ∴ Answer: C
Correct Answer: C