Find the unit vector parallel to the line  x-z=y=z

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:
\[
\boxed{\langle \frac{2}{\sqrt{6}}, \frac{1}{\sqrt{6}}, \frac{1}{\sqrt{6}} \rangle}
\]

---

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!

Find the unit vector parallel to the line x - z = y = z

Discover how to find the unit vector parallel to the line given by the equation x - z = y = z. This detailed solution includes key concepts from vector algebra and parametric equations, using normalization techniques to determine the unit vector. Perfect for students in Grades 10-12.

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