Find the angle between the vectors u and 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 u and 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 two vectors can be found using the dot product formula: $\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos \theta$. Calculating the dot product and magnitudes, we find $\cos \theta = -\frac{\sqrt{2}}{2}$, which corresponds to $\theta = 135^\circ$. However, considering the direction of vectors, the angle between them is $165^\circ$.

Find the Angle Between the Vectors U and V u=cos⁡(π3)i+sin⁡(π3)j\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j }u=cos(3π​)i+sin(3π​)j

Find the Angle Between the Vectors U and V $\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j }$