Find the angle between the vectors u and v. $\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j }$ , $v = \cos \left( \frac { 3 \pi } { 4 } \right) i + \sin \left( \frac { 3 \pi } { 4 } \right) j$
A) $40 ^ { \circ }$
B) $15 ^ { \circ }$
C) $75 ^ { \circ }$
D) $105 ^ { \circ }$
E) $70 ^ { \circ }$

Question

Find the angle between the vectors u and v. $\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j }$ , $v = \cos \left( \frac { 3 \pi } { 4 } \right) i + \sin \left( \frac { 3 \pi } { 4 } \right) j$ ​
A) $40 ^ { \circ }$
B) $15 ^ { \circ }$
C) $75 ^ { \circ }$
D) $105 ^ { \circ }$
E) $70 ^ { \circ }$

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 the vectors. Here, $\mathbf{u} = \cos\left(\frac{\pi}{3}\right)\mathbf{i} + \sin\left(\frac{\pi}{3}\right)\mathbf{j}$ and $\mathbf{v} = \cos\left(\frac{3\pi}{4}\right)\mathbf{i} + \sin\left(\frac{3\pi}{4}\right)\mathbf{j}$. Evaluating the trigonometric functions gives $\mathbf{u} = \frac{1}{2}\mathbf{i} + \frac{\sqrt{3}}{2}\mathbf{j}$ and $\mathbf{v} = -\frac{\sqrt{2}}{2}\mathbf{i} + \frac{\sqrt{2}}{2}\mathbf{j}$. The angle between these vectors can be calculated directly from their directions, leading to an angle of $75^\circ$.

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 $\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j }$