First slide

Dot product or scalar product of vectors

Question

$\vec{a} and \vec{b}$ are two vectors, such that $\vec{a} \cdot \vec{b} < 0 and | \vec{a} \cdot \vec{b} | = | \vec{a} \times \vec{b} |$ then the angle between vectors $\vec{a} and \vec{b}$ is

Moderate

A

$π$
B

$\frac{7 π}{4}$
C

$\frac{π}{4}$
D

$\frac{3 π}{4}$

Solution

$\begin{array}{l} | \vec{a} \cdot \vec{b} | = | \vec{a} \times \vec{b} | \\ \Rightarrow | \vec{a} | | \vec{b} | | \cos θ | = | \vec{a} | | \vec{b} | | \sin θ | \\ (where θ is the angle between \vec{a} and \vec{b}) \\ \Rightarrow ∣ \cos θ | = | \sin θ ∣ \\ \Rightarrow θ = \frac{π}{4} or \frac{3 π}{4} (as 0 \leq θ \leq π) \end{array}$
$But \vec{a} \cdot \vec{b} < 0, therefore θ = \frac{3 π}{4}$

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