Answer
The proof is below.
Work Step by Step
We need to prove $\triangle ABC$ is a right triangle. In order to 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.
Now, $\overrightarrow{AB} \cdot \overrightarrow{AC}=4(7)+(-2)(4)=20$
and
$\overrightarrow{AB} \cdot \overrightarrow{BC}=(4)(3)+(-2)(6)=0$
We can see that that the vectors $\overrightarrow{AB}$ and $ \overrightarrow{BC}$ forms a right angle. Thus, $\angle ABC=90^{\circ}$
