here are vectors v and w. find the magnitude and direction angle (θ) of v + w. round your final answer to…

Explanation:

Step1: Convert vectors to component - form

For vector $\vec{v}$ with magnitude $|\vec{v}| = 5$ and direction $\theta_v=180^{\circ}$, the components are $v_x=|\vec{v}|\cos\theta_v=5\cos180^{\circ}=- 5$ and $v_y = |\vec{v}|\sin\theta_v=5\sin180^{\circ}=0$. For vector $\vec{w}$ with magnitude $|\vec{w}| = 6$ and direction $\theta_w = 260^{\circ}$, $w_x=|\vec{w}|\cos\theta_w=6\cos260^{\circ}\approx - 0.10$ and $w_y=|\vec{w}|\sin\theta_w=6\sin260^{\circ}\approx - 5.99$.

Step2: Find the components of $\vec{v}+\vec{w}$

The $x$ - component of $\vec{v}+\vec{w}$ is $(\vec{v}+\vec{w})_x=v_x + w_x=-5+( - 0.10)=-5.10$. The $y$ - component of $\vec{v}+\vec{w}$ is $(\vec{v}+\vec{w})_y=v_y + w_y=0+( - 5.99)=-5.99$.

Step3: Calculate the magnitude of $\vec{v}+\vec{w}$

The magnitude $||\vec{v}+\vec{w}||=\sqrt{(\vec{v}+\vec{w})_x^2+(\vec{v}+\vec{w})_y^2}=\sqrt{(-5.10)^2+( - 5.99)^2}=\sqrt{26.01 + 35.88}=\sqrt{61.89}\approx7.9$.

Step4: Calculate the direction angle of $\vec{v}+\vec{w}$

The direction angle $\theta=\arctan\left(\frac{(\vec{v}+\vec{w})_y}{(\vec{v}+\vec{w})_x}\right)$. Since both $x$ and $y$ components are negative, the vector is in the third - quadrant. $\theta=\arctan\left(\frac{-5.99}{-5.10}\right)+180^{\circ}\approx49.6^{\circ}+180^{\circ}=229.6^{\circ}$.

Answer:

$||\vec{v}+\vec{w}||\approx7.9$ $\theta\approx229.6^{\circ}$