https://file.aiquickdraw.com/mathbot/file/1730071413191wdqzp187.jpg

To solve this problem, we need to follow these steps:

1. **Find vector \( c \) by calculating \( c = a - b \).**
   - Given:
     \[
     a = \langle 2, 10 \rangle
     \]
     \[
     b = \langle -5, -14 \rangle
     \]
   - Calculation:
     \[
     c = a - b = \langle 2 - (-5), 10 - (-14) \rangle = \langle 2 + 5, 10 + 14 \rangle = \langle 7, 24 \rangle
     \]

2. **Calculate the magnitude of vector \( c \).**
   - The magnitude \( |c| \) is calculated as:
     \[
     |c| = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25
     \]

3. **Find the direction angle \( \theta \) of vector \( c \).**
   - The direction angle \( \theta \) can be calculated using the formula:
     \[
     \theta = \tan^{-1}\left(\frac{24}{7}\right)
     \]
   - Calculating \( \theta \):
     \[
     \theta \approx \tan^{-1}(3.4286) \approx 73.7^\circ
     \]

### Conclusion
The magnitude of \( c \) is 25, and the direction angle \( \theta \) is approximately \( 73.7^\circ \).

### Answer
The correct answer is:
- \( |c| = 25, \theta = 73.7^\circ \)

Would you like more details or have any questions?

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Here are 5 related questions for further practice:

1. How would you calculate the magnitude and direction of a vector if it had components \( \langle x, y \rangle \)?
2. If \( b \) were \( \langle 5, -14 \rangle \) instead, what would be the new direction of \( c \)?
3. How do you determine the angle of a vector in the second quadrant?
4. What is the magnitude and angle of vector \( d = -c \)?
5. Can the angle of a vector be negative, and how would it affect its direction?

**Tip:** When calculating the direction of a vector, always check which quadrant it lies in to ensure the correct angle interpretation.

What is the magnitude and direction angle of vector c, given that c = a - b with a = ⟨2, 10⟩ and b = ⟨-5, -14⟩?

Learn how to find the magnitude and direction angle of vector c, derived from the subtraction of vectors a = ⟨2, 10⟩ and b = ⟨-5, -14⟩. This solution covers vector subtraction, magnitude calculation, and direction angle determination, suitable for Grades 10-12.

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