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.
$$\mathbf { u } = 4 \mathbf { i } + 2 \mathbf { j } - \mathbf { k } , \mathbf { v } = 7 \mathbf { i } - 7 \mathbf { j } + 4 \mathbf { k }$$
A) $11.6 ^ { \circ }$
B) $11.8 ^ { \circ }$
C) $78.4 ^ { \circ }$
D) $90 ^ { \circ }$
E) $78.2 ^ { \circ }$

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: $4*7 + 2*(-7) + (-1)*4 = 28 - 14 - 4 = 10$. Then, find the magnitudes: $|\mathbf{u}| = \sqrt{4^2 + 2^2 + (-1)^2} = \sqrt{16 + 4 + 1} = \sqrt{21}$ and $|\mathbf{v}| = \sqrt{7^2 + (-7)^2 + 4^2} = \sqrt{49 + 49 + 16} = \sqrt{114}$. Thus, $\cos(\theta) = \frac{10}{\sqrt{21} \sqrt{114}}$. Solving for $\theta$ gives $\theta \approx 78.2^\circ$.

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

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