10. Let $$ \boldsymbol{u}=\left(\begin{array}{c}-2 \ 5 \ 4\end{array} ight) $$, and $$ \boldsymbol{v}=\left(\begin{array}{c}6 \ -3 \ 0\end{array} ight) $$ Find, to the nearest degree, the measures of the angles between $$ \begin{array}{lll} ext { (a) } \boldsymbol{u} ext { and } \boldsymbol{v} & ext { (b) } \boldsymbol{u} ext { and }(\boldsymbol{u}+\boldsymbol{v}) & ext { (c) } \boldsymbol{v} ext { and }(\boldsymbol{u}+\boldsymbol{v})\end{array} $$

Question

10. Let $$ \boldsymbol{u}=\left(\begin{array}{c}-2 \ 5 \ 4\end{array}ight) $$, and $$ \boldsymbol{v}=\left(\begin{array}{c}6 \ -3 \ 0\end{array}ight) $$ Find, to the nearest degree, the measures of the angles between $$ \begin{array}{lll}	ext { (a) } \boldsymbol{u} 	ext { and } \boldsymbol{v} & 	ext { (b) } \boldsymbol{u} 	ext { and }(\boldsymbol{u}+\boldsymbol{v}) & 	ext { (c) } \boldsymbol{v} 	ext { and }(\boldsymbol{u}+\boldsymbol{v})\end{array} $$

UpStudy AI · Accepted Answer

$$
\textbf{(a)}
$$
We start with the formula for the angle $$\theta$$ between two vectors:
$$
\cos \theta = \frac{\textbf{u} \cdot \textbf{v}}{\|\textbf{u}\|\,\|\textbf{v}\|}
$$

For $$\textbf{u} = \begin{pmatrix} -2 \ 5 \ 4 \end{pmatrix}$$ and $$\textbf{v} = \begin{pmatrix} 6 \ -3 \ 0 \end{pmatrix}$$:

1. Compute the dot product:
$$
\textbf{u} \cdot \textbf{v} = (-2)(6) + (5)(-3) + (4)(0) = -12 - 15 = -27.
$$

2. Compute the magnitudes:
$$
\|\textbf{u}\| = \sqrt{(-2)^2 + 5^2 + 4^2} = \sqrt{4 + 25 + 16} = \sqrt{45} = 3\sqrt{5},
$$
$$
\|\textbf{v}\| = \sqrt{6^2 + (-3)^2 + 0^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}.
$$

3. Substitute into the cosine formula:
$$
\cos \theta = \frac{-27}{(3\sqrt{5})(3\sqrt{5})} = \frac{-27}{45} = -0.6.
$$

4. Solve for $$\theta$$:
$$
\theta = \arccos(-0.6) \approx 127^\circ.
$$

$$
\textbf{(b)}
$$
Now, find the angle between $$\textbf{u}$$ and $$\textbf{u} + \textbf{v}$$.

1. Compute $$\textbf{u} + \textbf{v}$$:
$$
\textbf{u} + \textbf{v} = \begin{pmatrix} -2+6 \ 5+(-3) \ 4+0 \end{pmatrix} = \begin{pmatrix} 4 \ 2 \ 4 \end{pmatrix}.
$$

2. Compute the dot product $$\textbf{u} \cdot (\textbf{u} + \textbf{v})$$:
$$
\textbf{u} \cdot (\textbf{u} + \textbf{v}) = (-2)(4) + (5)(2) + (4)(4) = -8 + 10 + 16 = 18.
$$

3. Compute the magnitude of $$\textbf{u} + \textbf{v}$$:
$$
\|\textbf{u} + \textbf{v}\| = \sqrt{4^2 + 2^2 + 4^2} = \sqrt{16 + 4 + 16} = \sqrt{36} = 6.
$$

4. Use the cosine formula:
$$
\cos \theta = \frac{18}{\|\textbf{u}\|\,\|\textbf{u} + \textbf{v}\|} = \frac{18}{(3\sqrt{5})(6)} = \frac{18}{18\sqrt{5}} = \frac{1}{\sqrt{5}} \approx 0.447.
$$

5. Solve for $$\theta$$:
$$
\theta = \arccos\left(\frac{1}{\sqrt{5}}\right) \approx 63^\circ.
$$

$$
\textbf{(c)}
$$
Next, find the angle between $$\textbf{v}$$ and $$\textbf{u} + \textbf{v}$$.

1. Compute the dot product $$\textbf{v} \cdot (\textbf{u} + \textbf{v})$$:
$$
\textbf{v} \cdot (\textbf{u} + \textbf{v}) = (6)(4) + (-3)(2) + (0)(4) = 24 - 6 = 18.
$$

2. Use the magnitudes already computed:
$$
\|\textbf{v}\| = 3\sqrt{5} \quad \text{and} \quad \|\textbf{u} + \textbf{v}\| = 6.
$$

3. Use the cosine formula:
$$
\cos \theta = \frac{18}{(3\sqrt{5})(6)} = \frac{18}{18\sqrt{5}} = \frac{1}{\sqrt{5}} \approx 0.447.
$$

4. Solve for $$\theta$$:
$$
\theta = \arccos\left(\frac{1}{\sqrt{5}}\right) \approx 63^\circ.
$$

$$
\boxed{\text{(a) } 127^\circ, \quad \text{(b) } 63^\circ, \quad \text{(c) } 63^\circ.}
$$
The angles are: - (a) 127 degrees - (b) 63 degrees - (c) 63 degrees

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