Draw point M(5,-3,4) and determine the length and direction…

Draw point $M(5,-3,4)$ and determine the length and direction of its radius vector.

Step 1: Radius vector

\[ \overrightarrow{OM} = 5\vec{i} - 3\vec{j} + 4\vec{k} \]

Step 2: Length

\[ r = \sqrt{5^2+(-3)^2+4^2} = \sqrt{25+9+16} = \sqrt{50} = 5\sqrt{2} \]

\[r = 5\sqrt{2} \approx 7.07\]

Two-step check: $d_{xy} = \sqrt{25+9} = \sqrt{34}$, then $r = \sqrt{34+16} = \sqrt{50}$ $\checkmark$

Step 3: Direction — direction cosines

\[ \cos\alpha = \frac{5}{5\sqrt{2}} = \frac{\sqrt{2}}{2} \;\Longrightarrow\; \alpha = 45^\circ \]

\[ \cos\beta = \frac{-3}{5\sqrt{2}} = -\frac{3\sqrt{2}}{10} \approx -0.4243 \;\Longrightarrow\; \beta \approx 115.1^\circ \]

\[ \cos\gamma = \frac{4}{5\sqrt{2}} = \frac{2\sqrt{2}}{5} \approx 0.5657 \;\Longrightarrow\; \gamma \approx 55.6^\circ \]

The angle $\beta$ is obtuse because $y \lt 0$: the vector points to the negative side of the $y$-axis.

Step 4: Check of the identity

\[ \cos^2\alpha+\cos^2\beta+\cos^2\gamma = \frac{1}{2}+\frac{9}{50}+\frac{16}{50} = \frac{25+9+16}{50} = 1 \;\checkmark \]

Step 5: Unit vector along $\overrightarrow{OM}$

\[\hat{r} = \frac{\sqrt{2}}{2}\,\vec{i} - \frac{3\sqrt{2}}{10}\,\vec{j} + \frac{2\sqrt{2}}{5}\,\vec{k}\]

Step 6: Drawing