3-33 할리데이 10판 솔루션 일반물리학

(a)$\vec r_a=\overrightarrow{(0,0,0)(a,a,a)}=?$ $$\vec r_a = a\hat i + a\hat j + a\hat k$$
(b)$\vec r_b=\overrightarrow{(a,0,0)(0,a,a)}=?$ $$\vec r_b = (-a)\hat i + a\hat j + a\hat k$$
(c)$\vec r_c=\overrightarrow{(0,a,0)(a,0,a)}=?$ $$\vec r_c = a\hat i + (-a)\hat j + a\hat k$$
(d)$\vec r_d=\overrightarrow{(a,a,0)(0,0,a)}=?$ $$\vec r_d = (-a)\hat i + (-a)\hat j + a\hat k$$
(e)$\phi_{\vec r \to \hat i,\hat j,\hat k}=?$ $$ \begin{aligned} \phi_{\vec r \to \hat i} =& \cos^{-1}\(\frac{\vec r \cdot \hat i}{ri}\)\\ =&\cos^{-1}\(\frac{(a\hat i + a\hat j + a\hat k)\cdot(\hat i)}{a\sqrt3\times1}\)\\ =&\cos^{-1}\(\frac{a\cdot1+a\cdot0+a\cdot0}{a\sqrt3}\)\\ =&\cos^{-1}\(\frac{a}{a\sqrt3}\)\\ =&\cos^{-1}\(\frac{1}{\sqrt3}\)\\ \approx&0.9553\ut{rad}\\ \end{aligned} $$ $$ \begin{aligned} \phi_{\vec r \to \hat j} =& \cos^{-1}\(\frac{\vec r \cdot \hat j}{rj}\)\\ =&\cos^{-1}\(\frac{(a\hat i + a\hat j + a\hat k)\cdot(\hat j)}{a\sqrt3\times1}\)\\ =&\cos^{-1}\(\frac{a\cdot0+a\cdot1+a\cdot0}{a\sqrt3}\)\\ =&\cos^{-1}\(\frac{a}{a\sqrt3}\)\\ =&\cos^{-1}\(\frac{1}{\sqrt3}\)\\ \approx&0.9553\ut{rad}\\ \end{aligned} $$ $$ \begin{aligned} \phi_{\vec r \to \hat k} =& \cos^{-1}\(\frac{\vec r \cdot \hat k}{rk}\)\\ =&\cos^{-1}\(\frac{(a\hat i + a\hat j + a\hat k)\cdot(\hat k)}{a\sqrt3\times1}\)\\ =&\cos^{-1}\(\frac{a\cdot0+a\cdot0+a\cdot1}{a\sqrt3}\)\\ =&\cos^{-1}\(\frac{a}{a\sqrt3}\)\\ =&\cos^{-1}\(\frac{1}{\sqrt3}\)\\ \approx&0.9553\ut{rad}\\ \end{aligned} $$
(f)$r=?$ $$ \begin{aligned} r=&\sqrt{a^2+a^2+a^2}\\ =&\sqrt{3a^2}\\ =&a\sqrt3\\ \approx&1.732a \end{aligned} $$