3-29 할리데이 10판 솔루션 일반물리학

(a),(b) $$\begin{cases} \vec w:(2.00\ut{cm},60^\circ)\\ \vec v:(2.00\ut{cm},90^\circ)\\ \vec i:(2.00\ut{cm},120^\circ)\\ \vec h:(2.00\ut{cm},90^\circ)\\ \end{cases} $$ $$ \begin{aligned} \vec w =& (2\cos60^\circ)\hat i + (2\sin60^\circ)\hat j\\ =&\hat i + \sqrt3\hat j \end{aligned} $$ $$ \begin{aligned} \vec v =& \vec h\\ =& (2\cos90^\circ)\hat i + (2\sin90^\circ)\hat j\\ =&2\hat j\\ \end{aligned} $$ $$ \begin{aligned} \vec i =& (2\cos120^\circ)\hat i + (2\sin120^\circ)\hat j\\ =&-\hat i + \sqrt3\hat j \end{aligned} $$ $$ \begin{aligned} \vec A =& \vec w + \vec v +\vec i + \vec h\\ =&\(4+2\sqrt3\)\hat j \end{aligned} $$
(a)$A=?$ $$ A=4+2\sqrt3$$
(b)$\theta_A=?$ $$ \theta_A = \frac{\pi}{2}$$
(c),(d) $$\begin{cases} \vec w:(2.00\ut{cm},60^\circ)\\ \vec v:(2.00\ut{cm},90^\circ)\\ \vec j:(2.00\ut{cm},60^\circ)\\ \vec p:(2.00\ut{cm},30^\circ)\\ \vec o:(2.00\ut{cm},0^\circ)\\ \end{cases} $$ $$ \begin{aligned} \vec j =& \vec w\\ =&\hat i + \sqrt3\hat j \end{aligned} $$ $$ \begin{aligned} \vec p =& (2\cos30^\circ)\hat i + (2\sin30^\circ)\hat j\\ =&\sqrt3\hat i + \hat j \end{aligned} $$ $$ \begin{aligned} \vec o =& (2\cos0^\circ)\hat i + (2\sin0^\circ)\hat j\\ =&2\hat i \end{aligned} $$ $$ \begin{aligned} \vec B =& \vec w+\vec v+\vec j+\vec p+\vec o\\ =&\(4+\sqrt3\)\hat i + \(3+2\sqrt3\)\hat j \end{aligned} $$
(a)$B=?$ $$ \begin{aligned} B=&\sqrt{\(4+\sqrt3\)^2+\(3+2\sqrt3\)^2}\\ =&2 \sqrt{5 \left(2+\sqrt{3}\right)}\\ \approx&8.64\ut{cm} \end{aligned} $$
(b)$\theta_B=?$ $$ \begin{aligned} \theta_B=&\tan^{-1}\(\frac{3+2\sqrt3}{4+\sqrt3}\)\\ =&\tan^{-1}\left(\frac{6+5\sqrt{3}}{13} \right)\\ =&0.845\ut{rad} \end{aligned} $$