10. Let \( \boldsymbol{u}=\left(\begin{array}{c}-2 \\ 5 \\ 4\end{array}\right) \), and \( \boldsymbol{v}=\left(\begin{array}{c}6 \\ -3 \\ 0\end{array}\right) \) Find, to the nearest degree, the measures of the angles between \( \begin{array}{lll}\text { (a) } \boldsymbol{u} \text { and } \boldsymbol{v} & \text { (b) } \boldsymbol{u} \text { and }(\boldsymbol{u}+\boldsymbol{v}) & \text { (c) } \boldsymbol{v} \text { and }(\boldsymbol{u}+\boldsymbol{v})\end{array} \)
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To find the angles between the given vectors, we can use the formula for the cosine of the angle between two vectors: \[ \cos(\theta) = \frac{\boldsymbol{u} \cdot \boldsymbol{v}}{\|\boldsymbol{u}\| \|\boldsymbol{v}\|} \] First, let's compute the necessary components: 1. **Dot product of \(\boldsymbol{u}\) and \(\boldsymbol{v}\)**: \[ \boldsymbol{u} \cdot \boldsymbol{v} = (-2)(6) + (5)(-3) + (4)(0) = -12 - 15 + 0 = -27 \] 2. **Magnitude of \(\boldsymbol{u}\)**: \[ \|\boldsymbol{u}\| = \sqrt{(-2)^2 + 5^2 + 4^2} = \sqrt{4 + 25 + 16} = \sqrt{45} = 3\sqrt{5} \] 3. **Magnitude of \(\boldsymbol{v}\)**: \[ \|\boldsymbol{v}\| = \sqrt{6^2 + (-3)^2 + 0^2} = \sqrt{36 + 9 + 0} = \sqrt{45} = 3\sqrt{5} \] (a) **Angle between \(\boldsymbol{u}\) and \(\boldsymbol{v}\)**: \[ \cos(\theta) = \frac{-27}{\|\boldsymbol{u}\| \|\boldsymbol{v}\|} = \frac{-27}{(3\sqrt{5})(3\sqrt{5})} = \frac{-27}{45} = -0.6 \] \[ \theta = \cos^{-1}(-0.6) \approx 127^\circ \] (b) **Finding \(\boldsymbol{u} + \boldsymbol{v}\)**: \[ \boldsymbol{u} + \boldsymbol{v} = \left(\begin{array}{c}-2 + 6 \\ 5 - 3 \\ 4 + 0\end{array}\right) = \left(\begin{array}{c}4 \\ 2 \\ 4\end{array}\right) \] **Magnitude of \((\boldsymbol{u} + \boldsymbol{v})\)**: \[ \|\boldsymbol{u} + \boldsymbol{v}\| = \sqrt{4^2 + 2^2 + 4^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6 \] **Dot product of \(\boldsymbol{u}\) and \((\boldsymbol{u} + \boldsymbol{v})\)**: \[ \boldsymbol{u} \cdot (\boldsymbol{u} + \boldsymbol{v}) = (-2)(4) + (5)(2) + (4)(4) = -8 + 10 + 16 = 18 \] **Cosine of angle**: \[ \cos(\theta) = \frac{18}{\|\boldsymbol{u}\| \|\boldsymbol{u} + \boldsymbol{v}\|} = \frac{18}{(3\sqrt{5})(6)} = \frac{18}{18\sqrt{5}} = \frac{1}{\sqrt{5}} \] \[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{5}}\right) \approx 63.4^\circ \] (c) **Dot product of \(\boldsymbol{v}\) and \((\boldsymbol{u} + \boldsymbol{v})\)**: \[ \boldsymbol{v} \cdot (\boldsymbol{u} + \boldsymbol{v}) = (6)(4) + (-3)(2) + (0)(4) = 24 - 6 + 0 = 18 \] **Cosine of angle**: \[ \cos(\theta) = \frac{18}{\|\boldsymbol{v}\| \|\boldsymbol{u} + \boldsymbol{v}\|} = \frac{18}{(3\sqrt{5})(6)} = \frac{18}{18\sqrt{5}} = \frac{1}{\sqrt{5}} \] \[ \theta = \cos^{-1}\left(\frac{1}{\sqrt{5}}\right) \approx 63.4^\circ \