91影视

Resolving vectors into their components using the vector algebra unknown vector can be found. The magnitude and the angle between the vector and +x axis can be calculated using the general equations given below for the magnitude and the angle.

Formulae

$$\begin{gathered} \stackrel{鈫�}{A}=\stackrel{鈫�}{A_x}+\stackrel{鈫�}{A_y}=A\cos 胃+A\sin 胃 \\ \stackrel{鈫�}{C}=\stackrel{鈫�}{C_x}+\stackrel{鈫�}{C_y}=C\cos 胃+C\sin 胃 \\ \stackrel{鈫�}{A}=\stackrel{鈫�}{B}+\stackrel{鈫�}{C} \\ {B}_x=C_x+A_x=C\cos 胃-A\cos 胃 \\ {B}_y=C_y+A_y=C\sin 胃-A\sin 胃 \\ {B}_y=\left|\stackrel{鈫�}{B}\right|=\sqrt{{\left(\stackrel{鈫�}{B_x}\right)}^2+{\left(\stackrel{鈫�}{B_y}\right)}^2} \\ 胃={\tan }^{-1}\left(\frac{\stackrel{鈫�}{B_y}}{B_x}\right) \\ \mathrm{Given}\mathrm{are}, \\ \mathrm{The}\mathrm{magnitude}\mathrm{of}\mathrm{vector}\stackrel{鈫�}{\mathrm{A}}=12\mathrm{m}\mathrm{and}\mathrm{angle}\mathrm{between}\stackrel{鈫�}{\mathrm{A}}\mathrm{and}+\mathrm{x}\mathrm{axis}\mathrm{is}40掳 \\ \mathrm{The}\mathrm{magnitude}\mathrm{of}\mathrm{vector}\stackrel{鈫�}{\mathrm{C}}=15\mathrm{m}\mathrm{and}\mathrm{angle}\mathrm{between}\stackrel{鈫�}{\mathrm{C}}\mathrm{and}+\mathrm{x}\mathrm{axis}\mathrm{is} \\ 180掳+20掳=200掳 \end{gathered}$$

$$\begin{gathered} {\stackrel{鈫�}{B}}_x={\stackrel{鈫�}{C}}_x-{\stackrel{鈫�}{A}}_x=\left(15.0\mathrm{m}\right)\cos \left(200\right)-\left(12.0\mathrm{m}\right)\cos \left(40\right) \\ {\stackrel{鈫�}{\mathrm{B}}}_{\mathrm{x}}=-23.3\mathrm{m} \\ {\stackrel{鈫�}{\mathrm{B}}}_{\mathrm{y}}={\stackrel{鈫�}{\mathrm{C}}}_{\mathrm{x}}-{\stackrel{鈫�}{\mathrm{A}}}_{\mathrm{x}}=\left(15.0\mathrm{m}\right)\sin \left(200\right)-\left(12.0\mathrm{m}\right)\sin \left(40\right) \\ {\stackrel{鈫�}{\mathrm{B}}}_{\mathrm{y}}=-12.8\mathrm{m} \end{gathered}$$

The magnitude of vector $\stackrel{鈫�}{B}$is

$\sqrt{{\left(-23.3\mathrm{m}\right)}^2+{\left(-12.8\mathrm{m}\right)}^2}=26.6\mathrm{m}$

The angle of vector $\stackrel{鈫�}{B}$ relative to +x axis is

${\tan }^{-1}\left(\frac{-12.8}{-23.3}\right)=28.9掳$

As vector $\stackrel{鈫�}{B}$is in third quadrant,

$180掳+28.9掳=209掳$