Find the angle, \( \alpha \), between the vectors.
\[
\begin{array}{l}
\overrightarrow{\mathrm{u}}=\langle-2,3\rangle \\
\overrightarrow{\mathrm{w}}=\langle 4,-2\rangle
\end{array}
\]

Answer

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Answer

\(\alpha =\cos^{-1}\left(\frac{-14}{\sqrt{13}\sqrt{20}}\right)\)

Steps

Step 1 ：\(\cos\alpha = \frac{\overrightarrow{u} \cdot \overrightarrow{w}}{||\overrightarrow{u}|| ||\overrightarrow{w}||}\)

Step 2 ：\(\cos\alpha = \frac{(-2)(4) + (3)(-2)}{\sqrt{(-2)^2 + 3^2} \sqrt{4^2 + (-2)^2}}\)

Step 3 ：\(\alpha =\cos^{-1}\left(\frac{-14}{\sqrt{13}\sqrt{20}}\right)\)