Question

FIGURE $P3.26$shows vectors $\overrightarrow{A}\&\overrightarrow{B}$. Find $\overrightarrow{D}=2\overrightarrow{A}+\overrightarrow{B}$. Write your answer in component form.

Short answer

The value of$\overrightarrow{D}=4\left(\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}\right)$.

Step-by-step solution

Step 1. Given information

The following diagram is given

Step 2. The vector calculation

The vectors are

$$\begin{gathered} \overrightarrow{A}=2\sin 15+2\cos 75 \\ =2\sqrt{2-\sqrt{3}} \\ \overrightarrow{B}=4\cos 15+4\sin 75 \\ =4\sqrt{2+\sqrt{3}} \end{gathered}$$

Hence,$$\begin{gathered} \overrightarrow{D}=2\overrightarrow{A}+\overrightarrow{B} \\ =4\sqrt{2+\sqrt{3}}+4\sqrt{2-\sqrt{3}} \\ =4\left(\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}}\right) \end{gathered}$$