Find the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$. Express your answer in degrees and round to the nearest tenth of a degree.
$$\mathbf { u } = - 5 \mathbf { i } - 6 \mathbf { j } - \mathbf { k } , \mathbf { v } = 6 \mathbf { i } + \mathbf { j } + \mathbf { k }$$
A) $37.3 ^ { \circ }$
B) $49.7 ^ { \circ }$
C) $52.7 ^ { \circ }$
D) $90 ^ { \circ }$
E) $139.7 ^ { \circ }$

Question

Quizplus · Accepted Answer

The angle between two vectors $\mathbf{u}$ and $\mathbf{v}$ can be found using the dot product formula: $\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos(\theta)$, where $\theta$ is the angle between the vectors. First, calculate the dot product: $(-5)(6) + (-6)(1) + (-1)(1) = -30 - 6 - 1 = -37$. Then, find the magnitudes of $\mathbf{u}$ and $\mathbf{v}$: $|\mathbf{u}| = \sqrt{(-5)^2 + (-6)^2 + (-1)^2} = \sqrt{25 + 36 + 1} = \sqrt{62}$, $|\mathbf{v}| = \sqrt{6^2 + 1^2 + 1^2} = \sqrt{36 + 1 + 1} = \sqrt{38}$. Using the dot product and magnitudes, solve for $\cos(\theta) = \frac{-37}{\sqrt{62} \sqrt{38}}$. Solving for $\theta$ gives an angle of approximately $139.7^\circ$.

Find the Angle Between the Vectors u\mathbf { u }u And v\mathbf { v }v

Find the Angle Between the Vectors $\mathbf { u }$ And $\mathbf { v }$