Let \hat{a} \textrm{ } and \textrm{ } \hat{b} be two unit vectors. If the vectors \overset{\rightarrow}{c} = \hat{a} + 2 \hat{b} \textrm{ }\textrm{ } and \textrm{ }\textrm{ } \overset{\rightarrow}{d} = 5 \hat{a} - 4 \hat{b} are perpendicular to each other, then the angle between \hat{a} \textrm{ } and \textrm{ } \hat{b} :

Engineering

Mathematics

Scalar Product of Two Vectors

Question

Let $\hat{a} and \hat{b}$ be two unit vectors. If the vectors $\vec{c} = \hat{a} + 2 \hat{b} and \vec{d} = 5 \hat{a} - 4 \hat{b}$ are perpendicular to each other, then the angle between $\hat{a} and \hat{b}$ :

$\frac{π}{3}$

$\frac{π}{6}$

$\frac{π}{4}$

$\frac{π}{2}$

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$\vec{c} \cdot \vec{d} = 0$

$(\vec{a} + 2 \vec{b}) \cdot (5 \vec{a} - 4 \vec{b}) = 0$

$5 | \vec{a} |^{2} + 6 \vec{a} \vec{b} - 8 | \vec{b} |^{2} = 0$

$5 + 6.1.1 cosθ - 8 = 0$

$cosθ = \frac{3}{6} = \frac{1}{2} \Rightarrow θ = \frac{π}{3}$

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