Question

Create an application different from any found in the text that requires a dot product.

Short answer

We can prove the law of cosines using dot product.

$$\begin{gathered} a^2=b^2+c^2-2bc\cos \alpha \\ {b}^2=a^2+c^2-2ac\cos \beta \\ {c}^2=a^2+b^2-2ab\cos \gamma \end{gathered}$$

Step-by-step solution

Step 1. An application that requires a dot product.

We can prove the law of cosines using dot product.

$$\begin{gathered} a^2=b^2+c^2-2bc\cos \alpha \\ {b}^2=a^2+c^2-2ac\cos \beta \\ {c}^2=a^2+b^2-2ab\cos \gamma \end{gathered}$$

Step 2. Proving the law of cosines using dot product.

Let three vectors represent the sides of the triangle.

Let

$$\begin{gathered} \left|\left|\overrightarrow{a}\right|\right|=a \\ \left|\left|\overrightarrow{b}\right|\right|=b \\ \left|\left|\overrightarrow{c}\right|\right|=c \end{gathered}$$

Let angle is $\alpha$between the $\overrightarrow{b}$and $\overrightarrow{c}$.

Apply triangle law of vector addition,

$$\begin{gathered} \overrightarrow{a}=\overrightarrow{b}+\overrightarrow{c} \\ \overrightarrow{a}·\overrightarrow{a}=\left(\overrightarrow{b}+\overrightarrow{c}\right)·\left(\overrightarrow{b}+\overrightarrow{c}\right) \\ {a}^2=\overrightarrow{b}·\overrightarrow{b}+\overrightarrow{b}·\overrightarrow{c}+\overrightarrow{c}·\overrightarrow{b}+\overrightarrow{c}·\overrightarrow{c} \\ {a}^2=b^2+c^2+2\overrightarrow{b}·\overrightarrow{c} \\ {a}^2=b^2+c^2+2\left|\left|\overrightarrow{b}\right|\right|\left|\left|\overrightarrow{c}\right|\right|\cos \alpha \\ {a}^2=b^2+c^2+2bc\cos \alpha \end{gathered}$$

Similarly we can prove

$$\begin{gathered} b^2=a^2+c^2-2ac\cos \beta \\ {c}^2=a^2+b^2-2ab\cos \gamma \end{gathered}$$