Find the angle between \(\over {OP}\) = (\(^{-3}_{-4}\)) and \(\over{OQ}\) = (\(^8_{-15}\))
To find the angle between two vectors, we can use the dot product formula:
\(\vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos\theta\)
where \(\vec{a}\) and \(\vec{b}\) are the two vectors, \(|\vec{a}|\) and \(|\vec{b}|\) are their magnitudes, and \(\theta\) is the angle between them.
First, we need to find the magnitudes of the two vectors:
\(|\overline{OP}| = \sqrt{(-3)^2 + (-4)^2} = 5\)
\(|\overline{OQ}| = \sqrt{8^2 + (-15)^2} = 17\)
Next, we need to find the dot product of the two vectors:
\(\overline{OP} \cdot \overline{OQ} = (-3)(8) + (-4)(-15) = 9\)
Now, we can substitute the values into the dot product formula to find the angle:
\(9 = 5 \cdot 17 \cdot \cos\theta\)
\(\cos\theta = \frac{9}{85}\)
\(\theta = \cos^{-1}\left(\frac{9}{85}\right)\)
Using a calculator, we get:
\(\theta \approx 71.4^\circ\)
Therefore, the angle between \(\overline{OP}\) and \(\overline{OQ}\) is approximately \(71.4^\circ\).