Find the magnitude and direction angle for the vector.

left angle 4 comma 9 right angle4, 9

To find the magnitude and direction angle of the vector \(\mathbf{v} = \langle 4, 9 \rangle\), we follow these steps:

1. **Magnitude**:
   The magnitude of a vector \(\mathbf{v} = \langle a, b \rangle\) is given by:
   \[
   \|\mathbf{v}\| = \sqrt{a^2 + b^2}
   \]
   For \(\mathbf{v} = \langle 4, 9 \rangle\):
   \[
   \|\mathbf{v}\| = \sqrt{4^2 + 9^2} = \sqrt{16 + 81} = \sqrt{97}
   \]

2. **Direction Angle**:
   The direction angle \(\theta\) of a vector is the angle it makes with the positive \(x\)-axis, and can be found using:
   \[
   \theta = \tan^{-1}\left(\frac{b}{a}\right)
   \]
   For \(\mathbf{v} = \langle 4, 9 \rangle\):
   \[
   \theta = \tan^{-1}\left(\frac{9}{4}\right)
   \]
   We need to compute this using a calculator:
   \[
   \theta \approx \tan^{-1}(2.25) \approx 66.04^\circ
   \]

Therefore, the magnitude and direction angle of the vector \(\mathbf{v} = \langle 4, 9 \rangle\) are:
- **Magnitude**: \(\sqrt{97} \approx 9.8\)
- **Direction Angle**: \(66.0^\circ\)

Would you like more details on any part of this solution, or do you have any other questions?

Here are some questions you might want to ask:
1. How is the direction angle of a vector determined?
2. What are the properties of vectors in two-dimensional space?
3. Can you explain the significance of the magnitude of a vector?
4. How do you add and subtract vectors?
5. What are some applications of vectors in real-world scenarios?

**Tip:** When finding the direction angle of a vector, always ensure your calculator is set to the correct mode (degrees or radians) based on the context of the problem.

Learn how to find the magnitude and direction angle of a vector \( \mathbf{v} = \langle 4, 9 \rangle \) with a detailed step-by-step solution. This problem covers key concepts in vectors, including magnitude and direction angles. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation