Find the direction angle of t= -4i+10j Round your answer to the nearest tenth of a degree.

To find the direction angle \(\theta\) of the vector \(\mathbf{t} = -4\mathbf{i} + 10\mathbf{j}\), you can use the following steps:

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

2. **Substitute the values**:
   \[
   \tan(\theta) = \frac{10}{-4} = -2.5
   \]

3. **Calculate the angle**:
   Find the arctan (inverse tangent) of \(-2.5\):
   \[
   \theta = \tan^{-1}(-2.5)
   \]

   Using a calculator:
   \[
   \theta \approx -68.2^\circ
   \]

4. **Adjust the angle**: Since the vector \(-4\mathbf{i} + 10\mathbf{j}\) is in the second quadrant (negative \(x\) and positive \(y\)), you need to add \(180^\circ\) to get the angle in standard position:
   \[
   \theta = 180^\circ - 68.2^\circ = 111.8^\circ
   \]

So, the direction angle \(\theta\) is approximately **111.8°**.

Would you like more details or have any questions?

Here are some related questions:
1. How do you find the direction angle of a vector given in three dimensions?
2. What is the significance of the direction angle in vector analysis?
3. How would the direction angle change if the vector was \(-4\mathbf{i} - 10\mathbf{j}\)?
4. Can the direction angle be negative, and how would you interpret it?
5. What other methods can you use to determine the direction of a vector?

**Tip**: When dealing with direction angles, always consider the quadrant in which the vector lies to adjust your final answer correctly.

Learn how to find the direction angle of vector -4i + 10j using trigonometric functions and inverse tangent. This problem involves calculating angles in vector analysis and is suitable for high school students.

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