https://file.aiquickdraw.com/mathbot/file/1726499975327g62imb4s.jpg

The problem in the image appears to state:

- **Given two vectors** \( \vec{a} + \vec{b} = 0 \) and the magnitudes \( |\vec{a}| = 5 \), \( |\vec{b}| = 12 \), find the **angle between** vectors \( \vec{a} \) and \( \vec{b} \).

### Solution:

1. From the equation \( \vec{a} + \vec{b} = 0 \), we know that the vectors \( \vec{a} \) and \( \vec{b} \) are opposite in direction, which implies that:
   \[
   \vec{b} = -\vec{a}
   \]
   
2. The **dot product formula** for two vectors is:
   \[
   \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos \theta
   \]
   where \( \theta \) is the angle between \( \vec{a} \) and \( \vec{b} \).

3. If \( \vec{b} = -\vec{a} \), then the angle between the two vectors must be **180 degrees**, because the two vectors point in exactly opposite directions.

Thus, the angle between \( \vec{a} \) and \( \vec{b} \) is **180°** or \( \pi \) radians.

---

Would you like a more detailed explanation, or do you have any specific questions? 

### Follow-up questions:
1. What is the dot product of two perpendicular vectors?
2. How does the magnitude of vectors affect their dot product?
3. Can two vectors with different magnitudes still be parallel?
4. What is the difference between the angle of \( 0^\circ \) and \( 180^\circ \) for vectors?
5. How would the result change if the magnitudes of \( \vec{a} \) and \( \vec{b} \) were equal?

### Tip:
Always remember that opposite vectors (i.e., those that add to zero) have an angle of \( 180^\circ \) between them!

If a + b = 0, |a| = 5, |b| = 12, find the angle between a and b.

This problem demonstrates how to find the angle between two vectors a and b, where a + b = 0, and the magnitudes of a and b are 5 and 12, respectively. Learn about the dot product and how opposite vectors result in an angle of 180°.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation