3-32 할리데이 10판 솔루션 일반물리학

(풀이자 주:x축이 아니라 z축을 나타내지 않았는데 오타로 추정됩니다. x축은 나타나있습니다.) $$\begin{cases} a:(4,0)\\ b:(3,\frac{\pi}{2})\\ c:(5,\pi+\tan^{-1}\(\frac{3}{4}\))\\ \end{cases}$$
(a)$\abs{\vec a \times \vec b}=?$ $$\begin{aligned} \Ans=&\abs{\vec a \times \vec b}\\ =& ab\sin\phi_{ab}\\ =& 4\cdot3\cdot1\\ =&12\\ \end{aligned}$$
(b)$\text{Direction of }\vec a \times \vec b=?$ $$+ z$$
(c)$\abs{\vec a \times \vec c}=?$ $$\begin{aligned} \Ans=&\abs{\vec a \times \vec c}\\ =& ac\sin\phi_{ac}\\ =& 4\cdot5\cdot\sin\bra{\pi-\tan^{-1}\(\frac{3}{4}\)}\\ =& 4\cdot5\cdot\sin\bra{\tan^{-1}\(\frac{3}{4}\)}\\ =& 20\cdot\(\frac{3}{5}\)\\ =&12 \end{aligned}$$
(d)$\text{Direction of }\vec a \times \vec c=?$ $$-z$$
(e)$\abs{\vec b \times \vec c}=?$ $$\begin{aligned} \Ans=&\abs{\vec b \times \vec c}\\ =& bc\sin\phi_{bc}\\ =& 3\cdot5\cdot\sin\bra{\pi+\tan^{-1}\(\frac{3}{4}\)-\frac{\pi}{2}}\\ =& 3\cdot5\cdot\sin\bra{\frac{\pi}{2}+\tan^{-1}\(\frac{3}{4}\)}\\ =& 3\cdot5\cdot\cos\bra{\tan^{-1}\(\frac{3}{4}\)}\\ =& 15\cdot\(\frac{4}{5}\)\\ =&12 \end{aligned}$$
(f)$\text{Direction of }\vec b \times \vec c=?$ $$+z$$