Contoh 7 Vektor

$\begin{array}{l}\\ 31.& \text{Jika}\overrightarrow{p}=\left(\begin{array}{c}2\\ -5\end{array}\right),\overrightarrow{q}=\left(\begin{array}{c}4\\ 3\end{array}\right),\text{maka }\\ & \text{proyeksi skalar ortogonal vektor}\overrightarrow{p}\\ & \text{pada}\overrightarrow{q}\text{adalah}....\\ & \text{a}.{\displaystyle \frac{3}{5}}\\ \\ & \text{b}.{\displaystyle \frac{7}{5}}\\ \\ & \text{c}.{\displaystyle \frac{8}{5}}\\ \\ & \text{d}.{\displaystyle \frac{9}{5}}\\ \\ & \text{e}.{\displaystyle 2}\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\left|\overrightarrow{r}\right|& ={\displaystyle \frac{\overrightarrow{p}.\overrightarrow{q}}{\left|\overrightarrow{q}\right|}}\\ & ={\displaystyle \frac{\left(\begin{array}{c}2\\ -5\end{array}\right).\left(\begin{array}{c}4\\ 3\end{array}\right)}{\sqrt{4^2+3^2}}}\\ & ={\displaystyle \frac{8+(-15)}{\sqrt{25}}}\\ & =\left|{\displaystyle \frac{-7}{5}}\right|\\ & ={\displaystyle \frac{7}{5}}\end{array}\end{array}$

$\begin{array}{l}\\ 32.& \text{Panjang Proyeksi vektor}\overrightarrow{a}=\left(\begin{array}{c}5\\ 1\end{array}\right)\\ & \text{pada}\overrightarrow{b}=\left(\begin{array}{c}0\\ 4\end{array}\right)\text{adalah}....\\ & \text{a}.{\displaystyle -1}\\ & \text{b}.-{\displaystyle \frac{1}{2}}\\ & \text{c}.1\\ & \text{d}.{\displaystyle 2}\\ & \text{e}.4\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\left|\overrightarrow{c}\right|& =\left|{\displaystyle \frac{\overrightarrow{a}.\overrightarrow{b}}{\left|\overrightarrow{b}\right|}}\right|\\ & =\left|{\displaystyle \frac{\left(\begin{array}{c}5\\ 1\end{array}\right)\bullet \left(\begin{array}{c}0\\ -4\end{array}\right)}{\left|\sqrt{0^2+(-4{)}^2}\right|}}\right|\\ & =\left|{\displaystyle \frac{0-4}{4}}\right|=|-1|=1\end{array}\end{array}$

$\begin{array}{l}\\ 33.& \text{Proyeksi vektor ortogonal}\overrightarrow{a}=\left(\begin{array}{c}2\\ -4\end{array}\right)\\ & \text{pada}\overrightarrow{b}=\left(\begin{array}{c}-1\\ 2\end{array}\right)\text{adalah}....\\ & \text{a}.\left(\begin{array}{c}2\\ -1\end{array}\right)\\ & \text{b}.\left(\begin{array}{c}2\\ -2\end{array}\right)\\ & \text{c}.\left(\begin{array}{c}2\\ -4\end{array}\right)\\ & \text{d}.\left(\begin{array}{c}-1\\ 2\end{array}\right)\\ & \text{e}.\left(\begin{array}{c}-2\\ 4\end{array}\right)\\ \\ & \text{Jawab}\\ & \begin{array}{rl}\overrightarrow{c}& =\left({\displaystyle \frac{\overrightarrow{a}\bullet \overrightarrow{b}}{{\left|\overrightarrow{b}\right|}^2}}\right).\overrightarrow{b}\\ & =\left({\displaystyle \frac{\left(\begin{array}{c}2\\ -4\end{array}\right)\bullet \left(\begin{array}{c}-1\\ 2\end{array}\right)}{(-1{)}^2+2^2}}\right).\left(\begin{array}{c}-1\\ 2\end{array}\right)\\ & =\left({\displaystyle \frac{-2-8}{1+4}}\right).\left(\begin{array}{c}-1\\ 2\end{array}\right)\\ & =-2\left(\begin{array}{c}-1\\ 2\end{array}\right)\\ & =\left(\begin{array}{c}2\\ -4\end{array}\right)\end{array}\end{array}$