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).