Find the component form of vector $\mathbf { u } + \mathbf { v } \text { given } \| \mathbf { u } \| = 2 \text { and } \| \mathbf { v } \| = 3 \text { and the angles }$ that $\mathbf { u }$ and $\mathbf { v }$ make with the positive $x$-axis are $\theta _ { \mathbf { u } } = 0 ^ { \circ }$ and $\theta _ { \mathbf { v } } = 45 ^ { \circ }$.
A) $\left( \frac { 4 + 3 \sqrt { 3 } } { 2 } , \frac { 9 \sqrt { 2 } } { 2 } \right)$
B)$\left( \frac { 2 + 3 \sqrt { 2 } } { 2 } , \frac { 9 \sqrt { 2 } } { 2 } \right)$
C)$\left\{ \frac { 22 + 9 \sqrt { 2 } } { 2 } , \frac { 6 \sqrt { 2 } } { 2 } \right\}$
D)$\left( \frac { 4 + 3 \sqrt { 2 } } { 2 } , \frac { 3 \sqrt { 2 } } { 2 } \right)$
E)$\left( \frac { 2 + 9 \sqrt { 2 } } { 2 } , \frac { 9 \sqrt { 2 } } { 2 } \right)$

Question

Quizplus · Accepted Answer

The component form of a vector given its magnitude and direction can be found using the formulas $x = r\cos(\theta)$ and $y = r\sin(\theta)$, where $r$ is the magnitude and $\theta$ is the angle the vector makes with the positive $x$-axis. For vector $\mathbf{u}$, since $\theta_{\mathbf{u}} = 0^\circ$, its components are $2\cos(0^\circ) = 2$ and $2\sin(0^\circ) = 0$, making $\mathbf{u} = (2, 0)$. For vector $\mathbf{v}$, with $\theta_{\mathbf{v}} = 45^\circ$, its components are $3\cos(45^\circ) = \frac{3\sqrt{2}}{2}$ and $3\sin(45^\circ) = \frac{3\sqrt{2}}{2}$, making $\mathbf{v} = \left(\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right)$. Adding $\mathbf{u}$ and $\mathbf{v}$ gives $\left(2 + \frac{3\sqrt{2}}{2}, 0 + \frac{3\sqrt{2}}{2}\right) = \left(\frac{4 + 3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right)$.

Find the Component Form of Vector u+v given ∥u∥=2 and ∥v∥=3 and the angles \mathbf { u } + \mathbf { v } \text { given } \| \mathbf { u } \| = 2 \text { and } \| \mathbf { v } \| = 3 \text { and the angles }u+v given ∥u∥=2 and ∥v∥=3 and the angles

Find the Component Form of Vector $\mathbf { u } + \mathbf { v } \text { given } \| \mathbf { u } \| = 2 \text { and } \| \mathbf { v } \| = 3 \text { and the angles }$