Let and are two unit vectors such that and are PERPENDICULAR. What is the angle between and ?

. Since they are perpendicular, their dot product is zero: . Expanding gives . Since they are unit vectors, . This simplifies to . Thus, , giving .

Let \vec{a} and \vec{b} are two unit vectors such that \vec{a}+2\vec{b} and 5\vec{a}-4\vec{b} are <strong>PERPENDICULAR</strong>. What is the angle between \vec{a} and \vec{b} ?

A. \frac{\pi}{6}
B. \frac{\pi}{4}
C. \frac{\pi}{3} ✓
D. \frac{\pi}{2}

Correct Answer: C. \frac{\pi}{3}

Explanation

Since they are perpendicular, their dot product is zero: (\vec{a}+2\vec{b}) \cdot (5\vec{a}-4\vec{b}) = 0. Expanding gives 5|\vec{a}|^2 + 10\vec{a}\cdot\vec{b} - 4\vec{a}\cdot\vec{b} - 8|\vec{b}|^2 = 0. Since they are unit vectors, |\vec{a}| = |\vec{b}| = 1. This simplifies to 5 + 6\vec{a}\cdot\vec{b} - 8 = 0 \implies 6\vec{a}\cdot\vec{b} = 3 \implies \vec{a}\cdot\vec{b} = \frac{1}{2}. Thus, \cos\theta = \frac{1}{2}, giving \theta = \frac{\pi}{3}.

Let \vec{a} and \vec{b} are two unit vectors such that \vec{a}+2\vec{b} and 5\vec{a}-4\vec{b} are <strong>PERPENDICULAR</strong>. What is the angle between \vec{a} and \vec{b} ?

Explanation

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