https://file.aiquickdraw.com/mathbot/file/17276281963614v9xes3s.jpg

To find the unit vector in the direction of **v** = (1, -2), follow these steps:

1. **Find the magnitude of vector v**:
   The magnitude \( \| \vec{v} \| \) of a vector \( \vec{v} = (x, y) \) is given by:
   \[
   \| \vec{v} \| = \sqrt{x^2 + y^2}
   \]
   So, for \( \vec{v} = (1, -2) \):
   \[
   \| \vec{v} \| = \sqrt{1^2 + (-2)^2} = \sqrt{1 + 4} = \sqrt{5}
   \]

2. **Divide each component of the vector by its magnitude**:
   The unit vector \( \hat{v} \) is calculated by dividing each component of \( \vec{v} \) by its magnitude:
   \[
   \hat{v} = \left( \frac{1}{\sqrt{5}}, \frac{-2}{\sqrt{5}} \right)
   \]
   You can rationalize the denominator, but the above form is sufficient unless otherwise required.

Thus, the unit vector in the direction of \( \vec{v} \) is:
\[
\left( \frac{1}{\sqrt{5}}, \frac{-2}{\sqrt{5}} \right)
\]

Would you like more details or help with another part?

Here are 5 related questions:
1. What is the significance of a unit vector in physics or geometry?
2. How can you generalize this process for any two-dimensional vector?
3. What happens if the vector components are all negative or zero?
4. How can unit vectors be used in determining direction in 3D space?
5. What does it mean if the magnitude of a vector is 1?

**Tip**: A unit vector always has a magnitude of 1 and retains the direction of the original vector.

Find the unit vector in the direction of v = (1, -2).

Learn how to find the unit vector in the direction of v = (1, -2) with a detailed step-by-step solution. This problem covers key vector concepts, including the magnitude and unit vector formulas. Suitable for high school 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