Use vectors to find the measure of the angle at vertex B of triangle ABC, when . Round answer to two decimal places.

Question

Use vectors to find the measure of the angle at vertex B of triangle ABC, when  . Round answer to two decimal places.

Quizplus · Accepted Answer

Let vector $\overrightarrow{BA} = \vec{a}$ and vector $\overrightarrow{BC} = \vec{c}$. 
Then, $\cos\angle ABC = \frac{\vec{a}\cdot\vec{c}}{\|\vec{a}\|\|\vec{c}\|}$ by the dot product formula. 
Substituting the given vectors, we have $\cos\angle ABC = \frac{-1\cdot1 + 2\cdot3}{\sqrt{2^2+(-1)^2}\sqrt{1^2+3^2}} = \frac{5}{\sqrt{10}\sqrt{10}} = \frac{1}{2}$. 
Therefore, $\angle ABC = \boxed{60^\circ}$, which is choice (B).

Use Vectors to Find the Measure of the Angle at Vertex