To find the unit vector of \( \mathbf{v} = 5 \mathbf{i} - 3 \mathbf{j} \), we follow these steps:

### Step 1: Find the magnitude of vector \( \mathbf{v} \)
The magnitude \( |\mathbf{v}| \) of the vector \( \mathbf{v} = 5 \mathbf{i} - 3 \mathbf{j} \) is given by the formula:

\[
|\mathbf{v}| = \sqrt{5^2 + (-3)^2} = \sqrt{25 + 9} = \sqrt{34}
\]

### Step 2: Calculate the unit vector
A unit vector is found by dividing the vector by its magnitude. The unit vector \( \mathbf{u} \) in the direction of \( \mathbf{v} \) is:

\[
\mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|} = \frac{5 \mathbf{i} - 3 \mathbf{j}}{\sqrt{34}}
\]

Thus, the unit vector is:

\[
\mathbf{u} = \frac{5}{\sqrt{34}} \mathbf{i} - \frac{3}{\sqrt{34}} \mathbf{j}
\]

If you'd like, I can also rationalize the denominators, but this is the simplest form.

Would you like any further details on the steps? Here are some related questions to explore:

1. How is a unit vector used in physics and geometry?
2. What is the significance of a vector's magnitude?
3. Can you find the unit vector of a 3D vector?
4. How does the concept of normalization relate to unit vectors?
5. What happens to the direction of a vector when converted to a unit vector?

**Tip**: The magnitude of a unit vector is always 1, regardless of its direction.

Learn how to find the unit vector of v = 5i - 3j using vector algebra. This problem involves calculating the magnitude and applying the unit vector formula, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation