Find the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$.
$\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j } , \quad \mathbf { v } = \cos \left( \frac { 5 \pi } { 4 } \right) \mathbf { i } + \sin \left( \frac { 5 \pi } { 4 } \right) \mathbf { j }$
A) $40 ^ { \circ }$
B) $75 ^ { \circ }$
C) $105 ^ { \circ }$
D) $165 ^ { \circ }$
E) $175 ^ { \circ }$

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 them. First, calculate the components of $\mathbf{u}$ and $\mathbf{v}$:- $\mathbf{u} = \left(\cos\left(\frac{\pi}{3}\right), \sin\left(\frac{\pi}{3}\right)\right) = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$- $\mathbf{v} = \left(\cos\left(\frac{5\pi}{4}\right), \sin\left(\frac{5\pi}{4}\right)\right) = \left(-\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2}\right)$The angle $\theta$ between $\mathbf{u}$ and $\mathbf{v}$ can be found by calculating the arccosine of the dot product of $\mathbf{u}$ and $\mathbf{v}$ divided by the product of their magnitudes. However, in this case, we can find the directional angles of both vectors and subtract them to find the angle between them. The directional angle of $\mathbf{u}$ is $\frac{\pi}{3}$ or 60 degrees, and the directional angle of $\mathbf{v}$ is $\frac{5\pi}{4}$ or 225 degrees. The difference between these angles is $225 - 60 = 165$ degrees, which is the angle between the two vectors.

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

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