Vektor 3 Dimensi

Cara 1:

Sebelum menentukan panjang AP, terlebih dahulu kita nyatakan vektor \(\overrightarrow{\text{AP}}\)

\(\overrightarrow{\text{AP}} = \overrightarrow{\text{AB}} + \overrightarrow{\text{BC}} + \overrightarrow{\text{CG}} + \overrightarrow{\text{GP}}\)

\(\overrightarrow{\text{AP}} = 10\textbf{j} \:-\: 10\textbf{i} + 10\textbf{k} + \frac{1}{2}\cdot \overrightarrow{\text{GH}}\)

\(\overrightarrow{\text{AP}} = 10\textbf{j} \:-\: 10\textbf{i} + 10\textbf{k} + \frac{1}{2}\cdot (-10)\textbf{j} \)

\(\overrightarrow{\text{AP}} = 10\textbf{j} \:-\: 10\textbf{i} + 10\textbf{k} \:-\:5\textbf{j} \)

\(\overrightarrow{\text{AP}} = -10\textbf{i} + 5\textbf{j} + 10\textbf{k} \)

Menghitung panjang AP

\(||\overrightarrow{\text{AP}}|| = \sqrt{(-10)^2 + 5^2 + 10^2}\)

\(||\overrightarrow{\text{AP}}|| = \sqrt{100 + 25 + 100}\)

\(||\overrightarrow{\text{AP}}|| = \sqrt{225}\)

\(||\overrightarrow{\text{AP}}|| = 15\text { satuan}\)

Cara 2:

Tentukan koordinat titik A dan P terlebih dahulu

Koordinat titik A adalah \((10, 0, 0)\) dan koordinat titik P adalah \((0, 5, 10)\)

\(\overrightarrow{\text{AP}} = \textbf{p}\:-\:\textbf{a}\)

\(\overrightarrow{\text{AP}} = \left(\begin{array}{c}0\\ 5\\10\end{array}\right)\:-\:\left(\begin{array}{c}10\\ 0\\0\end{array}\right) = \left(\begin{array}{c}-10\\ 5\\10\end{array}\right)\)

\(||\overrightarrow{\text{AP}}|| =\sqrt{(-10)^2 + 5^2 + 10^2}\)

\(||\overrightarrow{\text{AP}}|| =\sqrt{100 + 25 + 100}\)

\(||\overrightarrow{\text{AP}}|| =\sqrt{225}\)

\(||\overrightarrow{\text{AP}}|| = 15\text { satuan}\)

Cara 1:

Sebelum menentukan panjang PQ, terlebih dahulu kita nyatakan vektor \(\overrightarrow{\text{PQ}}\)

\(\overrightarrow{\text{PQ}} = \overrightarrow{\text{PF}} + \overrightarrow{\text{FB}} + \overrightarrow{\text{BC}} + \overrightarrow{\text{CQ}}\)

\(\overrightarrow{\text{PQ}} = \frac{3}{5}\cdot \overrightarrow{\text{EF}} + \overrightarrow{\text{FB}} + \overrightarrow{\text{BC}} + \frac{1}{5}\cdot \overrightarrow{\text{CG}}\)

\(\overrightarrow{\text{PQ}} = \frac{3}{5}\cdot 10\textbf{j} \:-\:10\textbf{k}\:-\:10\textbf{i} + \frac{1}{5}\cdot 10\textbf{k}\)

\(\overrightarrow{\text{PQ}} = 6\textbf{j} \:-\:10\textbf{k}\:-\:10\textbf{i} + 2\textbf{k}\)

\(\overrightarrow{\text{PQ}} = -10\textbf{i} + 6\textbf{j} \:-\:8\textbf{k}\)

Menghitung panjang PQ

\(||\overrightarrow{\text{PQ}}|| = \sqrt{(-10)^2 + 6^2 + (-8)^2}\)

\(||\overrightarrow{\text{PQ}}|| = \sqrt{100 + 36 + 64}\)

\(||\overrightarrow{\text{PQ}}|| = \sqrt{200}\)

\(||\overrightarrow{\text{PQ}}|| = 10\sqrt{2}\text { satuan}\)

Cara 2:

Tentukan koordinat titik P dan Q terlebih dahulu

Koordinat titik P adalah \((10, 4, 10)\) dan koordinat titik Q adalah \((0, 10, 2)\)

\(\overrightarrow{\text{PQ}} = \textbf{q}\:-\:\textbf{p}\)

\(\overrightarrow{\text{AP}} = \left(\begin{array}{c}0\\10\\2\end{array}\right)\:-\:\left(\begin{array}{c}10\\ 4\\10\end{array}\right) = \left(\begin{array}{c}-10\\6\\-8\end{array}\right)\)

\(||\overrightarrow{\text{PQ}}|| =\sqrt{(-10)^2 + 6^2 + (-8)^2}\)

\(||\overrightarrow{\text{PQ}}|| =\sqrt{100 + 36 + 64}\)

\(||\overrightarrow{\text{PQ}}|| =\sqrt{200}\)

\(||\overrightarrow{\text{PQ}}|| = 10\sqrt{2}\text { satuan}\)