Find the angle between \(\over {OP}\) = (\(^{-3}_{-4}\)) and \(\over{OQ}\) = (\(^8_{-15}\))
Question 1 Report
Find the angle between \(\over {OP}\) = (\(^{-3}_{-4}\)) and \(\over{OQ}\) = (\(^8_{-15}\))
Answer Details
\(\vec{OP}=\begin{pmatrix}-3\\-4\end{pmatrix},\ \vec{OQ}=\begin{pmatrix}8\\-15\end{pmatrix}\).
Dot product: \(\vec{OP}\cdot\vec{OQ}=(-3)(8)+(-4)(-15)=-24+60=36\).
Magnitudes: \(|\vec{OP}|=\sqrt{9+16}=5\), \(|\vec{OQ}|=\sqrt{64+225}=\sqrt{289}=17\).
\[\cos\theta=\frac{36}{5\times 17}=\frac{36}{85}=0.4235\Rightarrow \theta=\cos^{-1}(0.4235)\approx 65^\circ.\]The angle between the vectors is about \(65^\circ\).
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