Let u⃗ =[69],v⃗ =[−32],w⃗ =[10]u→=[69],v→=[−32],w→=[10] . Determine which type of angle (acute, obtuse, or right) is determined by u⃗ ⋅v⃗ ,u⃗ ⋅w⃗ ,v⃗ ⋅w⃗ u→⋅v→,u→⋅w→,v→⋅w→ .

To determine the type of angle between two vectors, we need to calculate their dot products. The dot product \( \vec{a} \cdot \vec{b} \) between two vectors \( \vec{a} \) and \( \vec{b} \) can be used to determine the angle \( \theta \) between them:

- If \( \vec{a} \cdot \vec{b} > 0 \), the angle is **acute**.
- If \( \vec{a} \cdot \vec{b} = 0 \), the angle is a **right** angle.
- If \( \vec{a} \cdot \vec{b} < 0 \), the angle is **obtuse**.

Given vectors:
- \( \vec{u} = [6, 9] \)
- \( \vec{v} = [-3, 2] \)
- \( \vec{w} = [1, 0] \)

Let's calculate each dot product:

1. **Dot product \( \vec{u} \cdot \vec{v} \):**
   \[
   \vec{u} \cdot \vec{v} = 6 \times (-3) + 9 \times 2 = -18 + 18 = 0
   \]
   Since the dot product is \( 0 \), the angle between \( \vec{u} \) and \( \vec{v} \) is a **right angle**.

2. **Dot product \( \vec{u} \cdot \vec{w} \):**
   \[
   \vec{u} \cdot \vec{w} = 6 \times 1 + 9 \times 0 = 6 + 0 = 6
   \]
   Since the dot product is positive (\( 6 \)), the angle between \( \vec{u} \) and \( \vec{w} \) is **acute**.

3. **Dot product \( \vec{v} \cdot \vec{w} \):**
   \[
   \vec{v} \cdot \vec{w} = -3 \times 1 + 2 \times 0 = -3 + 0 = -3
   \]
   Since the dot product is negative (\( -3 \)), the angle between \( \vec{v} \) and \( \vec{w} \) is **obtuse**.

### Summary:
- \( \vec{u} \cdot \vec{v} \): **Right angle**
- \( \vec{u} \cdot \vec{w} \): **Acute angle**
- \( \vec{v} \cdot \vec{w} \): **Obtuse angle**

Would you like more details or have any questions about these calculations?

Here are some related questions you might find interesting:
1. How do you find the magnitude of a vector?
2. What is the geometric interpretation of the dot product?
3. How can the dot product be used to project one vector onto another?
4. What is the cross product and how does it differ from the dot product?
5. How do you find the angle between two vectors using the dot product?

**Tip:** Remember, the dot product is not only useful for finding angles but also for checking orthogonality between vectors (dot product = 0).

Let u⃗ =[69], v⃗ =[−32], w⃗ =[10]. Determine which type of angle (acute, obtuse, or right) is determined by u⃗ ⋅v⃗ , u⃗ ⋅w⃗ , v⃗ ⋅w⃗.

Learn how to determine the type of angle (acute, obtuse, or right) between vectors using the dot product method. This problem involves calculating dot products for vectors u⃗ =[69], v⃗ =[−32], and w⃗ =[10] and interpreting the results to classify angles. Suitable for students in Grades 10-12.

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