솔루션피아

$$ \begin{cases} \vec r &= \(5.25\i+4.90\j+3.00\k\)\ut{m} \end{cases} $$ $$\ab{a}$$ $$ \begin{aligned} r &= \sqrt{5.25^2+4.9^2+3^2}\\ &=\frac{1}{20}\sqrt{24229}\ut{m}\\ &\approx 7.782833674183202\ut{m}\\ &\approx 7.78\ut{m}\\ \end{aligned} $$ $$\ab{b}$$ $$l\lt r ~?$$ $$\title{Impossible}$$ $$\ab{c}$$ $$l\gt r ~?$$ $$\title{Possible}$$ $$\ab{d}$$ $$l = r ~?$$ $$\title{Possible}$$ $$\ab{e}$$ $$\ve..

$$ \begin{cases} \vec A &= 7.00\ut{km}\j\\ B&=10.5\ut{km}\\ \theta_B&=60\degree\\ \end{cases} $$ $$ \begin{aligned} \vec B &=10.5\cos60\degree\i+10.5\sin60\degree\j\\ &=\frac{21}{4}\i+\frac{21}{4}\sqrt{3}\j \end{aligned} $$ $$ \begin{aligned} \vec C &=-\vec B+\vec A\\ &=-\(\frac{21}{4}\i+\frac{21}{4}\sqrt{3}\j\)+7\j\\ &=-\frac{21}{4}\i+\(7-\frac{21}{4}\sqrt{3}\)\j\\ \end{aligned} $$ $$\ab{a}$$ $..

$$ \begin{cases} \vec d &=-1.5\ut{m}\j \end{cases} $$ $$\ab{a}$$ $$ \begin{aligned} \abs{6.0\vec d} &= \abs{-9\j}\\ &=9 \end{aligned} $$ $$\ab{b}$$ $$\text{South}$$ $$\ab{c}$$ $$ \begin{aligned} \abs{-2.0\vec d} &= \abs{3\j}\\ &=3 \end{aligned} $$ $$\ab{b}$$ $$\text{North}$$

$$ \begin{cases} \vec a&=6.0\i+5.0\j\\ \vec b&=3.0\i+4.0\j\\ \end{cases} $$ $$\ab{a}$$ $$ \vec a\times \vec b=(a_yb_z-b_ya_z)\i+(a_zb_x-b_za_x)\j+(a_xb_y-b_xa_y)\k, $$ $$ \begin{aligned} \vec a \times \vec b &=\(6\i+5\j\)\times\(3\i+4\j\)\\ &=(6\times4-5\times3)\k\\ &=9\k \end{aligned} $$ $$\ab{b}$$ $$\vec a \cdot \vec b=a_xb_x+a_yb_y+a_zb_z,$$ $$ \begin{aligned} \vec a \cdot \vec b&=\(6\i+5\j\)..

$$ \begin{cases} \text{East}:+\i\\ \text{North}:+\j\\ \text{up}:+\k\\ \end{cases} $$ $$\ab{a}$$ $$ \begin{aligned} \Ans&=\(-\j\)\times\(-\i\)\\ &=(-1)(-1)\(\j\times\i\)\\ &=\(\j\times\i\)\\ &=(-1)\(\i\times\j\)\\ &=(-1)\(\k\)\\ &=-\k\\ &\text{Down} \end{aligned} $$ $$\ab{b}$$ $$ \begin{aligned} \Ans&=\(-\k\)\times\(-\j\)\\ &=(-1)(-1)\(\k\times\j\)\\ &=\(\k\times\j\)\\ &=(-1)\(\j\times\k\)\\ &=(-..

$$ \begin{cases} \vec A + \vec B &=1.0\i+6.0\j\\ \vec A - \vec B &=-5.0\i+2.0\j\\ \end{cases} $$ $$ \begin{aligned} 2\vec A&=-4.0\i+8.0\j\\ \vec A&=-2.0\i+4.0\j\\ \end{aligned} $$

$$ \begin{cases} \vec d_1 &= \(-2.0\i+3.0\j+4.0\k\)\ut{m}\\ \vec d_2 &= \(-2.0\i-4.0\j+3.0\k\)\ut{m}\\ \vec d_3 &= \(2.0\i+3.0\j+2.5\k\)\ut{m}\\ \end{cases} $$ $$\ab{a}$$ $$\vec a \cdot \vec b=a_xb_x+a_yb_y+a_zb_z,$$ $$ \begin{aligned} \Ans&=\vec d_1\cdot\(\vec d_2+\vec d_3\)\\ &=\vec d_1\cdot\bra{\(-2\i-4\j+3\k\)+\(2\i+3\j+2.5\k\)}\\ &=\(-2\i+3\j+4\k\)\cdot\(-\j+5.5\k\)\\ &=(-2)\cdot0+3\cdot(-1..

$$ \begin{cases} \vec a &= 8\i\\ \vec b &= 6\j\\ \vec c &= -8\i-6\j\\ \end{cases} $$ $$\ab{a}$$ $$ \begin{aligned} \vec a\cdot\vec b&=8\cdot0+0\cdot6\\ &=0 \end{aligned} $$ $$\ab{b}$$ $$ \begin{aligned} \vec a\cdot\vec c&=8\cdot(-8)+0\cdot(-6)\\ &=-64 \end{aligned} $$ $$\ab{c}$$ $$ \begin{aligned} \vec b\cdot\vec c&=0\cdot(-8)+6\cdot(-6)\\ &=-36 \end{aligned} $$

$$ \begin{cases} \vec A &= 0\\ \vec B &= 55\ut{km}\i\\ r_1&=24\ut{km}, \theta_1=-25\degree\\ \vec r_2 &= 8.0\ut{km}\j \end{cases} $$ $$ \begin{aligned} \vec C &= \(\vec A + \vec r_1 + \vec r_2\)-\vec B\\ &=\(0 + 24\cos(-25\degree)\i+24\sin(-25\degree) + 8\j\)-55\i\\ &=(24\cos25\degree-55)\i+(8-24\sin25\degree)\j\\ \end{aligned} $$ $$ \begin{aligned} C &= \sqrt{(24\cos25\degree-55)^2+(8-24\sin25\..

$$ \begin{cases} \vec a &= a\i\\ \vec b &= -b\j\\ d&>0 \end{cases} $$ $$\ab{a}$$ $$ \begin{aligned} \frac{\vec b}{d}&=\frac{-b\j}{d}\\ &=-\frac{b}{d}\j\\ \end{aligned} $$ $$-\frac{b}{d}\lt0$$ $$\therefore -y$$ $$\ab{b}$$ $$ \begin{aligned} \frac{\vec b}{-d}&=\frac{-b\j}{-d}\\ &=\frac{b}{d}\j\\ \end{aligned} $$ $$\frac{b}{d}\gt0$$ $$\therefore +y$$ $$\ab{c}$$ $$ \begin{aligned} \vec a \cdot \vec ..