α
β
γ
\begin{align*}
\cos\alpha &= \frac{\mathbf{v}\cdot\mathbf{i}}{|\mathbf{v}||\mathbf{i}|} = \frac{v_1}{|\mathbf{v}|} \\
&= \frac{2}{\sqrt{2^2+3^2+2^2}} = \frac{2}{\sqrt{17}} \\ \\
\alpha &= \cos^{-1}\left(\frac{2}{\sqrt{17}}\right) \approx 1.064\mbox{ rad}
\end{align*}
\begin{align*}
\cos\beta &= \frac{\mathbf{v}\cdot\mathbf{j}}{|\mathbf{v}||\mathbf{j}|} = \frac{v_2}{|\mathbf{v}|} \\
&= \frac{3}{\sqrt{2^2+3^2+2^2}} = \frac{3}{\sqrt{17}} \\ \\
\beta &= \cos^{-1}\left(\frac{3}{\sqrt{17}}\right) \approx 0.756\mbox{ rad}
\end{align*}
\begin{align*}
\cos\gamma &= \frac{\mathbf{v}\cdot\mathbf{k}}{|\mathbf{v}||\mathbf{k}|} = \frac{v_3}{|\mathbf{v}|} \\
&= \frac{2}{\sqrt{2^2+3^2+2^2}} = \frac{2}{\sqrt{17}} \\ \\
\gamma &= \cos^{-1}\left(\frac{2}{\sqrt{17}}\right) \approx 1.064\mbox{ rad}
\end{align*}