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?

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.