Find the direction angle of w= (-9,-9) *Round* *your answer to the nearest tenth of a* *degree.*

To find the direction angle \(\theta\) of the vector \(\mathbf{w} = (-9, -9)\), follow these steps:

1. **Formula**: The direction angle \(\theta\) is given by:
   \[
   \tan(\theta) = \frac{y}{x}
   \]
   where \(x\) and \(y\) are the components of the vector.

2. **Substitute the values**:
   \[
   \tan(\theta) = \frac{-9}{-9} = 1
   \]

3. **Calculate the angle**:
   Find the arctan (inverse tangent) of \(1\):
   \[
   \theta = \tan^{-1}(1)
   \]

   Using a calculator:
   \[
   \theta \approx 45.0^\circ
   \]

4. **Adjust the angle**: Since the vector \((-9, -9)\) is in the third quadrant (both \(x\) and \(y\) are negative), you need to add \(180^\circ\) to get the angle in standard position:
   \[
   \theta = 180^\circ + 45.0^\circ = 225.0^\circ
   \]

So, the direction angle \(\theta\) is **225.0°**.

Would you like more details or have any questions?

Here are some related questions:
1. How does the direction angle change when moving from one quadrant to another?
2. What is the difference between the direction angle and the angle of elevation?
3. How would you calculate the magnitude of the vector \(\mathbf{w}\)?
4. What would the direction angle be if \(\mathbf{w} = (9, 9)\)?
5. How do you find the direction angle in three-dimensional space?

**Tip**: Always consider the signs of the vector components to determine the correct quadrant before finalizing the direction angle.

Learn how to find the direction angle of vector w = (-9, -9) using trigonometric principles. Step-by-step solution provided, rounding the answer to the nearest tenth of a degree. Suitable for students in Grades 10-12.

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