## Linear Algebra: A Modern Introduction

We need to prove that $\triangle ABC$ is a right triangle. In order to do this, we will have to find a right angle. When the dot product for two vectors is zero, they form a right angle. Let us take three vectors $\overrightarrow{AB},\overrightarrow{AC},\overrightarrow{BC}$ and find their dot product. If we find one of the dot products is 0, we can conclude that there is a right angle. Now, $\overrightarrow{AB} \cdot \overrightarrow{AC}=-4(1)+(1)(1)+(-1)(-3)=-4+4=0$ We can see that that the vectors $\overrightarrow{AB}$ and $\overrightarrow{AC}$ form a right angle. Thus, $\angle BAC=90^{\circ}$