Given points A(2,2,0) and B(0,-2,5) . Construct vector…

Given points $A(2,2,0)$ and $B(0,-2,5)$. Construct vector $\overrightarrow{AB}$ and determine its magnitude, direction and unit vector (ort).

Step 1: Component form

By Problem 28: $\overrightarrow{AB} = (0-2)\vec{i}+(-2-2)\vec{j}+(5-0)\vec{k}$.

\[\overrightarrow{AB} = -2\vec{i}-4\vec{j}+5\vec{k}\]

Step 2: Magnitude

\[ |\overrightarrow{AB}| = \sqrt{4+16+25} = \sqrt{45} = 3\sqrt{5} \approx 6.708 \]

Two-step check: $d_{xy} = \sqrt{4+16} = 2\sqrt{5}$, then $\sqrt{20+25} = 3\sqrt5$ $\checkmark$

Step 3: Direction cosines

\[ \cos\alpha = \frac{-2}{3\sqrt{5}} = -\frac{2\sqrt{5}}{15} \approx -0.298 \;\Longrightarrow\; \alpha \approx 107.3^\circ \]

\[ \cos\beta = \frac{-4}{3\sqrt{5}} = -\frac{4\sqrt{5}}{15} \approx -0.596 \;\Longrightarrow\; \beta \approx 126.6^\circ \]

\[ \cos\gamma = \frac{5}{3\sqrt{5}} = \frac{\sqrt{5}}{3} \approx 0.745 \;\Longrightarrow\; \gamma \approx 41.8^\circ \]

Check: $\dfrac{4+16+25}{45} = 1$ $\checkmark$. The angles $\alpha$ and $\beta$ are obtuse because the $x$- and $y$-components are negative.

Step 4: Unit vector (ort)

\[\hat{r}_{AB} = \frac{\overrightarrow{AB}}{|\overrightarrow{AB}|} = -\frac{2\sqrt{5}}{15}\,\vec{i} - \frac{4\sqrt{5}}{15}\,\vec{j} + \frac{\sqrt{5}}{3}\,\vec{k}\]

$|\hat{r}_{AB}|^2 = \frac{4}{45}+\frac{16}{45}+\frac{25}{45} = 1$ $\checkmark$

Step 5: Construction

From $A$, go $5$ up, then $-2$ along $x$, then $-4$ along $y$: \[A(2,2,0) \xrightarrow{+5\vec{k}} (2,2,5) \xrightarrow{-2\vec{i}} (0,2,5) \xrightarrow{-4\vec{j}} B(0,-2,5)\]