In each part, find two unit vectors in 2-space that satisfy the stated condition. (c) Perpendicular to the line y = −5x + 1


To find two unit vectors in 2-space that are **perpendicular** to the line \( y = -5x + 1 \), we first need to understand the slope of the given line.

### Step 1: Identify the slope of the line
The equation of the line is given as \( y = -5x + 1 \). The slope of this line is the coefficient of \(x\), which is \( m = -5 \).

### Step 2: Find the slope of the perpendicular line
The slope of a line perpendicular to another is the **negative reciprocal** of the original slope. So, the slope of the line perpendicular to \( y = -5x + 1 \) is:
\[
m_{\perp} = \frac{1}{5}
\]

### Step 3: Determine the direction vector
The direction vector of a line with slope \( \frac{1}{5} \) can be written as \( \langle 1, 5 \rangle \) since the slope \( \frac{1}{5} \) is the ratio of the vertical component to the horizontal component.

### Step 4: Normalize the vector to get the unit vector
To find a **unit vector**, we need to normalize \( \langle 1, 5 \rangle \).

The magnitude of the vector \( \langle 1, 5 \rangle \) is:
\[
\| \langle 1, 5 \rangle \| = \sqrt{1^2 + 5^2} = \sqrt{1 + 25} = \sqrt{26}
\]

The unit vector in the direction of \( \langle 1, 5 \rangle \) is:
\[
\mathbf{u}_1 = \left\langle \frac{1}{\sqrt{26}}, \frac{5}{\sqrt{26}} \right\rangle
\]

### Step 5: Find the opposite unit vector
Since the problem asks for **two** unit vectors, one in the opposite direction is simply the negative of \( \mathbf{u}_1 \):
\[
\mathbf{u}_2 = \left\langle -\frac{1}{\sqrt{26}}, -\frac{5}{\sqrt{26}} \right\rangle
\]

### Final Answer:
The two unit vectors perpendicular to the line \( y = -5x + 1 \) are:
\[
\mathbf{u}_1 = \left\langle \frac{1}{\sqrt{26}}, \frac{5}{\sqrt{26}} \right\rangle \quad \text{and} \quad \mathbf{u}_2 = \left\langle -\frac{1}{\sqrt{26}}, -\frac{5}{\sqrt{26}} \right\rangle
\]

Would you like more details on any step or have further questions?

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Here are 5 related questions:
1. How do you find the unit vector of a given vector?
2. What is the significance of the negative reciprocal when finding perpendicular slopes?
3. Can two vectors with different magnitudes still be perpendicular?
4. How do you verify that two vectors are perpendicular?
5. How do you convert a line equation to vector form?

**Tip:** When working with vectors, always ensure to normalize them if you're asked to find unit vectors. This ensures their magnitude is 1.

In each part, find two unit vectors in 2-space that satisfy the stated condition. (c) Perpendicular to the line y = −5x + 1

Learn how to find two unit vectors in 2-space that are perpendicular to the line y = −5x + 1. This detailed solution explains the concept of perpendicularity, unit vectors, and vector normalization using step-by-step calculations. Suitable for students in Grades 9-12.

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