솔루션피아

$$\begin{cases} R&=22.0\ut{cm}\\ \vec P_1&=0\\ \vec P_2&=\cfrac{2\pi R}{2}\i+2Rj \end{cases}$$ $$\begin{aligned} \Delta \vec P &= \vec P_2\\ &= \pi R\i+2R\j\\ \end{aligned}$$ $$\ab{a}$$ $$\begin{aligned} P&=\sqrt{(\pi R)^2+(2R)^2}\\ &=R\sqrt{\pi^2+4}\\ &=22\sqrt{\pi^2+4}\ut{cm}\\ &\approx 81.93221912121781\ut{cm}\\ &\approx 81.9\ut{cm} \end{aligned}$$ $$\ab{b}$$ $$ \begin{aligned} \theta_P..

$$\begin{cases} \vec r_a&=(90.0\ut{km})\i\\ \vec r_b&=(60.0\ut{km})\j\\ \end{cases}$$ $$\begin{aligned} \vec r_s &= -\vec r_b+\vec r_a\\ &= 90\i-60\j \end{aligned}$$ $$\ab{a}$$ $$\begin{aligned} r_s &= \sqrt{90^2+60^2}\\ &= 30\sqrt{13}\ut{km}\\ &\approx 108.1665382639197\ut{km}\\ &\approx 108\ut{km}\\ \end{aligned}$$ $$\ab{b}$$ $$ \begin{aligned} \theta_{x+}&=\tan^{-1}\frac{60}{90}\\ &\app..

$$\begin{cases} \vec r_a&=800\i\ut{m}\\ \vec r_b&=700\j\ut{m}\\ \vec r_c&=-300\k\ut{m}\\ \end{cases}$$ $$\ab{a}$$ $$\begin{aligned} \Sigma \vec r &= \(800\ut{m}\)\i+\(700\ut{m}\)\j+\(-300\ut{m}\)\k\\ \end{aligned}$$ $$\ab{b}$$ $$ \begin{aligned} \Ans &= \abs{-\Sigma \vec r}\\ &=\abs{\Sigma \vec r}\\ &=\sqrt{800^2+700^2+(-300)^2}\\ &=100\sqrt{122}\ut{m}\\ &\approx 1104.536101718726\ut{m}\\ &\..

$$\begin{cases} a&=3.90\\ b&=2.70\\ \phi_{ab}&=27.0\degree\\ \end{cases}$$ $$\begin{aligned} \Ans&=\abs{\vec a\times (\vec b \times \vec a)}\\ &=\abs{\vec a\times \vec c}\\ &=ac ~(\because \vec a \perp \vec c)\\ &=a\cdot\abs{\vec b \times \vec a}\\ &=a\cdot(ba\sin\phi)\\ &=a^2b\sin\phi\\ &=(3.9)^2(2.7)\sin27\degree\\ &\approx 18.64402785280397\\ &\approx 18.6\\ \end{aligned}$$

$$\begin{cases} E=5.0\ut{m}, \theta_E=+0.9\ut{rad}\\ F=6.0\ut{m}, \theta_F=-75.0\degree\\ G=7.00\ut{m},\theta_G=+1.2\ut{rad}\\ H=4.00\ut{m}, \theta_H=-210\degree\\ \end{cases}$$ $$\begin{cases} \vec E =& 5\cos0.9\i+5\sin0.9\j\\ \vec F =& 6\cos(-75.0\degree)\i+6\sin(-75.0\degree)\j\\ \vec G =& 7\cos1.2\i+7\sin1.2\j\\ \vec H =& 4\cos(-210\degree)\i+4\sin(-210\degree)\j\\ \end{cases}$$ $$\ab{a}..

$$\begin{cases} A&=8.00\\ \theta_A&=130\degree\\ \vec B&=-4.31\i-6.60\j\\ \end{cases}$$ $$\ab{a}$$ $$\begin{aligned} \theta_{Ay-}&=\theta_A+90\degree\\ &= 220\degree\\ &=\frac{11}{9}\pi\ut{rad}\\ &\approx3.839724354387525\ut{rad}\\ &\approx3.84\ut{rad}\\ \end{aligned}$$ $$\ab{b}$$ $$\begin{aligned} \vec A = 8\cos130\degree\i+8\sin130\degree\j\\ \end{aligned}$$ $$ \begin{aligned} \vec C&=\v..

$$\begin{cases} \vec a&=(90.0\ut{km})\j\\ \vec b&=(35.0\ut{km})\i\\ \end{cases}$$ $$\begin{aligned} \vec c &= \vec a + (-\vec b)\\ &=90\j-35\i\\ \end{aligned}$$ $$\ab{a}$$ $$\begin{aligned} c&=\abs{\vec c}\\ &= \sqrt{(-35)^2+90^2}\ut{km}\\ &= 5 \sqrt{373}\ut{km}\\ &\approx 96.56603957913984\ut{km}\\ &\approx 96.6\ut{km}\\ \end{aligned}$$ $$\ab{b}$$ $$ \begin{aligned} \theta_c&=\tan^{-1}\(\..

$$\begin{cases} \vec d_1 &= 3\i-2\j+4\k\\ \vec d_2 &= -7\i+2\j-\k\\ \end{cases}$$ $$\begin{aligned} \Ans&=\(\vec d_1 + \vec d_2\)\cdot(\vec d_1\times 16\vec d_2)\\ &=\(-4\i+3\k\)\cdot\bra{\(3\i-2\j+4\k\)\times\(-112\i+32\j-16\k\)}\\ &=\(-4\i+3\k\)\cdot\(-96\i-400\j-128\k\)\\ &=0 \end{aligned}$$

$$\begin{cases} q&=2.0\\ \vec v&= 2.0\i+4.0\j+6.0\k\\ \vec F&=4.0\i-32\j+20\k\\ B_x&=B_y \end{cases}$$ $$\vec a\times \vec b=(a_yb_z-b_ya_z)\i+(a_zb_x-b_za_x)\j+(a_xb_y-b_xa_y)\k,$$ $$\begin{aligned} q\vec v\times \vec B&=2\(2\i+4\j+6\k\)\times\(B_x\i+B_x\j+B_z\k\)\\ &=\(4\i+8\j+12\k\)\times\(B_x\i+B_x\j+B_z\k\)\\ &=\(-12B_x+8B_z\)\i+\(12B_x-4B_z\)\j+\(-4B_x\)\k\\ \end{aligned}$$ $$ \begin..

$$\begin{cases} a&=9.5\\ b&=7.5\\ \phi_{ab}&=55\degree\\ \end{cases}$$ $$\ab{a}$$ $$\begin{aligned} \vec a \cdot \vec b&=ab\cos\phi\\ &=(9.5)(7.5)\cos55\degree\\ &\approx 40.86732109001203\\ &\approx 41\\ \end{aligned}$$ $$\ab{b}$$ $$\begin{aligned} \abs{\vec a \times \vec b}&=ab\sin\phi\\ &=(9.5)(7.5)\sin55\degree\\ &\approx 58.36458315559067\\ &\approx 58\\ \end{aligned}$$