To find two unit vectors in 2-space that are parallel to the line $y = 3x + 2$, we first need to identify the direction of this line. The equation of the line is in the form:

$y = mx + b$

where $m$ is the slope of the line. Here, the slope $m = 3$, which means the line has a direction vector proportional to $\langle 1, 3 \rangle$, since a slope of 3 corresponds to a rise of 3 units for every run of 1 unit.

Step 1: Find the direction vector

The line's direction vector is $\langle 1, 3 \rangle$. To find a unit vector, we first normalize this vector.

The magnitude (length) of $\langle 1, 3 \rangle$ is:

$\text{Magnitude} = \sqrt{1^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10}$

Step 2: Normalize the vector

To make this a unit vector, we divide each component of the vector $\langle 1, 3 \rangle$ by its magnitude $\sqrt{10}$:

$\text{Unit vector} = \left\langle \frac{1}{\sqrt{10}}, \frac{3}{\sqrt{10}} \right\rangle$

This is one unit vector in the direction of the line.

Step 3: Find the opposite unit vector

Since the line extends in both directions, there is also a unit vector in the opposite direction. This is simply the negative of the previous unit vector:

$\text{Opposite unit vector} = \left\langle \frac{-1}{\sqrt{10}}, \frac{-3}{\sqrt{10}} \right\rangle$

Final Answer:

The two unit vectors parallel to the line $y = 3x + 2$ are:

1. $\left\langle \frac{1}{\sqrt{10}}, \frac{3}{\sqrt{10}} \right\rangle$
2. $\left\langle \frac{-1}{\sqrt{10}}, \frac{-3}{\sqrt{10}} \right\rangle$

Would you like more details or have any questions?

Here are five related questions:

1. How do you find the direction vector from any general line equation?
2. What is the significance of unit vectors in physics and engineering?
3. How do you find the magnitude of a vector in 3D space?
4. What is the difference between direction vectors and unit vectors?
5. How do you normalize a vector in higher dimensions?

Tip: Unit vectors are essential because they provide direction without changing the magnitude of vectors. They are often used to simplify vector calculations in physics and geometry.