Construct vector r=overrightarrowOM=i+3j+5k and determine…

Construct vector $\vec{r}=\overrightarrow{OM}=\vec{i}+3\vec{j}+5\vec{k}$ and determine its magnitude and direction.

Explanation of Method

Constructing $\vec r=\vec i+3\vec j+5\vec k$ and determining its magnitude and direction.

Comparing with $x\vec{i}+y\vec{j}+z\vec{k}$: $x = 1,\ y = 3,\ z = 5$. So $\overrightarrow{OM}$ goes from $O(0,0,0)$ to $M(1,3,5)$.

\[ r = \sqrt{1^2+3^2+5^2} = \sqrt{1+9+25} = \sqrt{35} \]

\[r = \sqrt{35} \approx 5.916\]

Two-step check: $d_{xy} = \sqrt{1+9} = \sqrt{10}$, then $r = \sqrt{10+25} = \sqrt{35}$ $\checkmark$

\[ \cos\alpha = \frac{1}{\sqrt{35}} \approx 0.1690 \;\Longrightarrow\; \alpha \approx 80.3^\circ \]

\[ \cos\beta = \frac{3}{\sqrt{35}} \approx 0.5071 \;\Longrightarrow\; \beta \approx 59.5^\circ \]

\[ \cos\gamma = \frac{5}{\sqrt{35}} \approx 0.8452 \;\Longrightarrow\; \gamma \approx 32.3^\circ \]

Check: $\cos^2\alpha+\cos^2\beta+\cos^2\gamma = \dfrac{1+9+25}{35} = 1$ $\checkmark$

\[\hat{r} = \frac{1}{\sqrt{35}}\vec{i}+\frac{3}{\sqrt{35}}\vec{j}+\frac{5}{\sqrt{35}}\vec{k}\]

\[ O \xrightarrow{\ \vec{i}\ } (1,0,0) \xrightarrow{\ 3\vec{j}\ } M'(1,3,0) \xrightarrow{\ 5\vec{k}\ } M(1,3,5) \]

$\overrightarrow{OM}$ is the space diagonal of the box with edges $1$, $3$ and $5$ along the axes.

$\alpha$ $\beta$ $\gamma$ $O$ $M(1,3,5)$ $M'(1,3,0)$ $\vec i$ $3\vec j$ $5\vec k$

$\vec r$ is the space diagonal of the box with edges $1,\ 3,\ 5$; path $O\to(1,0,0)\to M'\to M$.