https://file.aiquickdraw.com/mathbot/file/1730430215559q41zbdcm.jpg

Let’s analyze each of these properties one by one.

### Statement A
**\( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \)**

This statement is about the dot product. The dot product of two vectors is commutative, meaning \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \) is **True**.

### Statement B
**\( \vec{a} \times \vec{b} = \vec{b} \times \vec{a} \)**

This statement is about the cross product. The cross product of two vectors is **not commutative**; in fact, \( \vec{a} \times \vec{b} = -(\vec{b} \times \vec{a}) \). So, this statement is **False**.

### Statement C
**\( \vec{u} \cdot \vec{u} = \| \vec{u} \| \)**

This statement is a test of the relationship between the dot product and the magnitude of a vector. The dot product \( \vec{u} \cdot \vec{u} \) gives \( \| \vec{u} \|^2 \), not \( \| \vec{u} \| \). Therefore, this statement is **False**.

### Statement D
**\( \| \vec{u} \times \vec{v} \| = \| \vec{u} \| \| \vec{v} \| \sin(\theta) \)**

This statement is about the magnitude of the cross product. The magnitude of the cross product \( \| \vec{u} \times \vec{v} \| \) is defined as \( \| \vec{u} \| \| \vec{v} \| \sin(\theta) \), where \( \theta \) is the angle between \( \vec{u} \) and \( \vec{v} \). So, this statement is **True**.

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### Summary of Answers:
1. **Statement A:** True
2. **Statement B:** False
3. **Statement C:** False
4. **Statement D:** True

Would you like more details or have any questions?

### Related Questions
1. What is the geometric interpretation of the dot product?
2. How can we calculate the cross product of two vectors?
3. What is the significance of the angle in the cross product's magnitude formula?
4. How does the commutativity of the dot product differ from that of the cross product?
5. How do we interpret \( \vec{u} \cdot \vec{u} \) in vector mathematics?

### Tip
Remember, the cross product is only defined in three dimensions, whereas the dot product can be calculated in any dimension.

For each property, indicate whether it's True or False: 
A) \( \vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{a} \) 
B) \( \vec{a} \times \vec{b} = \vec{b} \times \vec{a} \) 
C) \( \vec{u} \cdot \vec{u} = \| \vec{u} \| \) 
D) \( \| \vec{u} \times \vec{v} \| = \| \vec{u} \| \| \vec{v} \| \sin(\theta) \)

Evaluate the properties of dot and cross products for vectors. This problem explores key vector operations, including the commutativity of dot product, non-commutativity of cross product, and magnitude of vector products. Ideal for high school students in Grades 10-12.

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