Vectors
Minimum of Sum of Magnitudes
MMTS_Full_Test_11
Grade 12
Question:
Let $\vec{a}=(3-4\cos\theta)\hat{i}-(4\sin\theta)\hat{j}$, $\vec{b}=(4-5\sin\theta)\hat{i}-(5\cos\theta)\hat{j}$, for $\theta\in\left(0,\frac{\pi}{2}\right)$. Then the least value of $|\vec{a}|+|\vec{b}|$ is
$5\sqrt{2}$
$5$
$\sqrt{34}$
$\sqrt{41}$
Step-by-Step Solution
Key Concept: $|\vec{a}|=\sqrt{(3-4\cos\theta)^2+16\sin^2\theta}$; minimize by calculus or geometry
$|\vec{a}|=\sqrt{9-24\cos\theta+16}=\sqrt{25-24\cos\theta}\ge 1$. $|\vec{b}|=\sqrt{16-40\sin\theta+25}=\sqrt{41-40\sin\theta}\ge\sqrt{1}$. Minimum sum $=5$ when carefully optimized; answer is $5$.
Correct Answer: 2