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.
$$\mathbf { u } = - 6 \mathbf { i } + 9 \mathbf { j } + 2 \mathbf { k } , \mathbf { v } = - 3 \mathbf { i } - 7 \mathbf { j } - 4 \mathbf { k }$$
A) $29.3 ^ { \circ }$
B) $34.1 ^ { \circ }$
C) $60.7 ^ { \circ }$
D) $90 ^ { \circ }$
E) $124.1 ^ { \circ }$

Question

Quizplus · Accepted Answer

The angle $\theta$ between two vectors $\mathbf{u}$ and $\mathbf{v}$ can be found using the dot product formula: $\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}| |\mathbf{v}| \cos(\theta)$. First, calculate the dot product: $(-6)(-3) + (9)(-7) + (2)(-4) = 18 - 63 - 8 = -53$. Then, find the magnitudes of $\mathbf{u}$ and $\mathbf{v}$: $|\mathbf{u}| = \sqrt{(-6)^2 + 9^2 + 2^2} = \sqrt{36 + 81 + 4} = \sqrt{121} = 11$, $|\mathbf{v}| = \sqrt{(-3)^2 + (-7)^2 + (-4)^2} = \sqrt{9 + 49 + 16} = \sqrt{74}$. Substitute these into the formula and solve for $\theta$: $\cos(\theta) = \frac{-53}{11\sqrt{74}}$, $\theta = \cos^{-1}\left(\frac{-53}{11\sqrt{74}}\right) \approx 124.1^\circ$.

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

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