Find the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$.
$$\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j } , \mathbf { v } = \cos \left( \frac { 5 \pi } { 4 } \right) \mathbf { i } + \sin \left( \frac { 5 \pi } { 4 } \right) \mathbf { j }$$
A) $40 ^ { \circ }$

B) $75 ^ { \circ }$

C) $105 ^ { \circ }$

D) $165 ^ { \circ }$

E) $175 ^ { \circ }$

Question

Find the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$.
$$\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j } , \mathbf { v } = \cos \left( \frac { 5 \pi } { 4 } \right) \mathbf { i } + \sin \left( \frac { 5 \pi } { 4 } \right) \mathbf { j }$$
A) $40 ^ { \circ }$

B) $75 ^ { \circ }$

C) $105 ^ { \circ }$

D) $165 ^ { \circ }$

E) $175 ^ { \circ }$

Quizplus · Accepted Answer

The angle between vectors $\mathbf{u}$ and $\mathbf{v}$ is found using the dot product formula. The dot product $\mathbf{u} \cdot \mathbf{v} = \cos(\frac{\pi}{3})\cos(\frac{5\pi}{4}) + \sin(\frac{\pi}{3})\sin(\frac{5\pi}{4})$ results in $-\frac{\sqrt{6}}{4} - \frac{\sqrt{2}}{4}$. The magnitudes of $\mathbf{u}$ and $\mathbf{v}$ are both 1. Thus, $\cos(\theta) = -\frac{\sqrt{6} + \sqrt{2}}{4}$, leading to $\theta = 165^\circ$.

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

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