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The image contains several mathematical statements related to vector algebra and mixed products (also known as scalar triple products). Here’s a breakdown of the content:

1. **Mixed Product Identity**:
   \[
   \langle \vec{a} + \vec{b}; \vec{c}; \vec{d} \rangle = \langle \vec{c}; \vec{d}; \vec{a} \rangle + \langle \vec{b}; \vec{c}; \vec{d} \rangle
   \]
   This identity shows a linear property of the mixed product.

2. **Mixed Product in 3-Dimensional Space**:
   In 3D space \( V^3 \), the mixed product of basis vectors is equal to zero. This likely refers to the fact that if vectors form a basis, and one of the vectors is a linear combination of the others, the scalar triple product is zero.

3. **Dot Product of Opposite Vectors**:
   For two non-zero, oppositely directed vectors \( \vec{a} \) and \( \vec{b} \), the dot product is:
   \[
   (\vec{a}, \vec{b}) = -|\vec{a}| \cdot |\vec{b}|
   \]
   This is consistent with the definition of the dot product when the vectors point in opposite directions.

4. **Volume of a Tetrahedron**:
   The volume of a tetrahedron formed by three vectors is equal to the absolute value of their mixed product:
   \[
   V_{\text{tetrahedron}} = |\langle \vec{a}; \vec{b}; \vec{c} \rangle|
   \]

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

Here are 5 related questions to expand your understanding:
1. How is the mixed product defined for three vectors, and what geometric significance does it hold?
2. Why is the scalar triple product equal to zero for dependent vectors?
3. How does the direction of two vectors affect the sign of their dot product?
4. What is the relationship between the scalar triple product and the volume of a parallelepiped?
5. How can you interpret the scalar triple product using determinants?

**Tip:** The scalar triple product \(\langle \vec{a}; \vec{b}; \vec{c} \rangle\) gives the signed volume of the parallelepiped spanned by three vectors.

Identify the correct statements regarding vector algebra and scalar triple products, including mixed product identities, properties in 3D space, the dot product of opposite vectors, and the volume of a tetrahedron.

Explore vector algebra concepts including scalar triple products, mixed product identities, the dot product of opposite vectors, and the formula for the volume of a tetrahedron. Ideal for advanced high school or undergraduate students.

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