3-42 할리데이 10판 솔루션 일반물리학

$$\begin{cases} \vec d_1 = (4.0 \ut{m})\i + (5.0 \ut{m})\j\\ \vec d_2= (-3.0 \ut{m})\i + (4.0 \ut{m})\j\\ \end{cases}$$
(a) $\vec d_1 \times \vec d_2=?$ $$\begin{aligned} \vec d_1 \times \vec d_2=&\begin{vmatrix} \i & \j & \k\\ 4&5&0\\ -3&4&0\\ \end{vmatrix}\\ =&\begin{vmatrix} 5&0\\ 4&0\\ \end{vmatrix}\i+ \begin{vmatrix} 0&4\\ 0&-3\\ \end{vmatrix}\j+ \begin{vmatrix} 4&5\\ -3&4\\ \end{vmatrix}\k\\ =&31\k \end{aligned}$$
(b) $\vec d_1 \cdot \vec d_2=?$ $$\vec d_1 \cdot \vec d_2=4\cdot(-3)+5\cdot4=8.0$$
(c) $(\vec d_1+\vec d_2)\cdot \vec d_2=?$ $$\begin{aligned} (\vec d_1+\vec d_2)\cdot \vec d_2=&\bra{(4-3)\i+(5+4)\j}\cdot\vec d_2\\ =&1\cdot(-3)+9\cdot4\\ =&33 \end{aligned}$$
(d) $\vec d_1$ 의 $\vec d_2$방향 성분 $$d_1=\sqrt{4^2+5^2}=\sqrt{41}$$ $$d_2=\sqrt{(-3)^2+4^2}=5$$ $$\begin{aligned} \cos\phi_{d_1d_2}=&\frac{d_1\cdot d_2}{d_1d_2}\\ =&\frac{8}{5\sqrt{41}} \end{aligned}$$ $$\begin{aligned} d_1\cos\phi_{d_1d_2}=&\sqrt{41}\cdot\frac{8}{5\sqrt{41}}\\ =&\frac{8}{5}\\=&1.6\\ \end{aligned}$$