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.