here are vectors ????⃗ and ????⃗. find the magnitude and direction angle (θ) of ????⃗ + ????⃗. round your…

Explanation:

Step1: Convert vectors to component - form

For vector $\vec{v}$ with magnitude $|\vec{v}| = 8$ and direction angle $\theta_v=100^{\circ}$: $v_x=8\cos(100^{\circ})\approx8\times(- 0.1736)\approx - 1.39$ $v_y=8\sin(100^{\circ})\approx8\times0.9848\approx7.88$

For vector $\vec{w}$ with magnitude $|\vec{w}| = 3$ and direction angle $\theta_w = 210^{\circ}$: $w_x=3\cos(210^{\circ})=3\times(-\frac{\sqrt{3}}{2})\approx3\times(-0.8660)\approx - 2.60$ $w_y=3\sin(210^{\circ})=3\times(-\frac{1}{2})=-1.50$

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

$(\vec{v}+\vec{w})_x=v_x + w_x\approx-1.39-2.60=-3.99\approx - 4.0$ $(\vec{v}+\vec{w})_y=v_y + w_y\approx7.88-1.50 = 6.38\approx6.4$

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

$|\vec{v}+\vec{w}|=\sqrt{(\vec{v}+\vec{w})_x^2+(\vec{v}+\vec{w})_y^2}=\sqrt{(-4.0)^2+(6.4)^2}=\sqrt{16 + 40.96}=\sqrt{56.96}\approx7.5$

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

$\theta=\arctan(\frac{(\vec{v}+\vec{w})_y}{(\vec{v}+\vec{w})_x})+180^{\circ}$ (since $(\vec{v}+\vec{w})_x<0$) $\theta=\arctan(\frac{6.4}{-4.0})+180^{\circ}$ $\theta=\arctan(-1.6)+180^{\circ}\approx - 58.0^{\circ}+180^{\circ}=122.0^{\circ}$

Answer:

Magnitude: $7.5$, Direction angle: $122.0^{\circ}$