We are given the equation of a line in the form $x - z = y = z$. Let's break this down and rewrite it in parametric form, which will help us find the direction of the line, and from there, the unit vector parallel to it.

Step 1: Rewrite the equation of the line

The given equation is: $x - z = y = z$ This can be split into two parts:

1. $y = z$
2. $x = 2z$ (because $x - z = z$)

Let $z = t$ where $t$ is a parameter. Now we can express the coordinates $x$, $y$, and $z$ in terms of $t$: $x = 2t, \quad y = t, \quad z = t$

Step 2: Parametric form of the line

So, the parametric form of the line is: $(x, y, z) = (2t, t, t)$ The direction vector of the line is obtained by considering the change in $x$, $y$, and $z$ with respect to the parameter $t$, which gives us the vector: $\vec{v} = \langle 2, 1, 1 \rangle$

Step 3: Find the unit vector

To find the unit vector parallel to the line, we first need the magnitude of the direction vector $\vec{v} = \langle 2, 1, 1 \rangle$: $|\vec{v}| = \sqrt{2^2 + 1^2 + 1^2} = \sqrt{4 + 1 + 1} = \sqrt{6}$

Now, the unit vector $\hat{v}$ is found by dividing each component of $\vec{v}$ by $|\vec{v}|$: $\hat{v} = \frac{1}{\sqrt{6}} \langle 2, 1, 1 \rangle = \langle \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}} \rangle$

To simplify, multiply the components by $\frac{\sqrt{6}}{6}$ to get the unit vector in a simpler form: $\hat{v} = \langle \frac{\sqrt{6}}{3}, \frac{\sqrt{6}}{6}, \frac{\sqrt{6}}{6} \rangle$

Thus, the unit vector parallel to the given line is: $\boxed{\langle \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}} \rangle}$

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Would you like more details on any of the steps? Here are some related questions you might explore:

1. How do you find the parametric form of more complex lines?
2. What is the importance of unit vectors in vector mathematics?
3. Can you explain how to normalize a vector in more detail?
4. What are some applications of unit vectors in physics or engineering?
5. How do direction cosines relate to unit vectors?

Tip: Always double-check vector magnitudes when normalizing to ensure accuracy!