
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Express the dot product $\vec{r}_{1} \cdot \vec{r}_{2}$ in coordinates if $\vec{r}_{1}=x_{1} \vec{i}+y_{1} \vec{j}+z_{1} \vec{k}$ and $\vec{r}_{2}=x_{2} \vec{i}+y_{2} \vec{j}+z_{2} \vec{k}$. Calculate $\cos \varphi$, where $\varphi$ is the angle between the vectors $\vec{r}_{1}$ and $\vec{r}_{2}$. Write the conditions in coordinates for the collinearity and orthogonality of the vectors $\vec{r}_{1}$ and $\vec{r}_{2}$. Which formula expresses the distance between the points whose coordinates are $( x_{1}, y_{1}, z_{1} )$ and $( x_{2}, y_{2}, z_{2} )$?

Present the answer in LaTex format: \boxed{Your answer}
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