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.