Let v = ⟨-5, 12⟩. What is the approximate direction angle of v?

23°

67°

113°

157°

To find the direction angle \( \theta \) of a vector \( v = \langle -5, 12 \rangle \), we can use the formula:

\[
\theta = \tan^{-1} \left( \frac{y}{x} \right)
\]

where \( x = -5 \) and \( y = 12 \).

1. **Calculate \( \tan^{-1} \left( \frac{12}{-5} \right) \):**

   \[
   \theta = \tan^{-1} \left( \frac{12}{-5} \right) \approx \tan^{-1}(-2.4)
   \]

   Using a calculator, we find:

   \[
   \theta \approx -67.38^\circ
   \]

2. **Adjust for the correct quadrant:**

   Since the vector \( \langle -5, 12 \rangle \) is in the second quadrant (negative \( x \)-value and positive \( y \)-value), we add 180° to the angle:

   \[
   \theta = -67.38^\circ + 180^\circ \approx 112.62^\circ
   \]

Thus, the approximate direction angle is:

\[
\theta \approx 113^\circ
\]

**Answer: 113°**

Would you like further details on this calculation or have additional questions?

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Here are some related questions for further practice:

1. What is the direction angle of a vector in the fourth quadrant?
2. How would the direction angle change if the vector were \( \langle 5, -12 \rangle \)?
3. Can you find the magnitude of \( v = \langle -5, 12 \rangle \)?
4. What is the direction angle of \( \langle 12, -5 \rangle \)?
5. How do we interpret a negative angle in vector direction?

**Tip:** Remember, if the vector lies in the second or third quadrant, add 180° to the inverse tangent result to find the correct direction angle.

Let v = ⟨-5, 12⟩. What is the approximate direction angle of v? 

23°
67°
113°
157°

Find the approximate direction angle of the vector v = ⟨-5, 12⟩ with a step-by-step solution using inverse tangent calculations. This problem explores direction angles and vector analysis concepts, ideal for high school students in Grades 10-12.

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