To prove the vector identity $\lambda (\mathbf{B} \times \mathbf{C}) = \mathbf{B}(\mathbf{C} \times \mathbf{A}) = \mathbf{C}(\mathbf{A} \times \mathbf{B})$, we will break it into parts and establish the relationship between these vectors.

Step 1: Define the scalar λ in terms of vectors

Assume that the scalar $\lambda$ is related to the scalar triple product, which represents the volume of the parallelepiped formed by three vectors $\mathbf{A}, \mathbf{B}, \mathbf{C}$. The scalar triple product is defined as:

$\lambda = \mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$

Step 2: Prove the first identity

Starting with $\lambda (\mathbf{B} \times \mathbf{C})$:

$\lambda (\mathbf{B} \times \mathbf{C}) = [\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})](\mathbf{B} \times \mathbf{C})$

This expression involves the scalar triple product $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$, which is a scalar multiplied by the vector $\mathbf{B} \times \mathbf{C}$. This term represents the volume of a parallelepiped formed by vectors $\mathbf{A}, \mathbf{B}, \mathbf{C}$, scaled by the vector $\mathbf{B} \times \mathbf{C}$.

Step 3: Prove the second identity

Now, consider the term $\mathbf{B} (\mathbf{C} \times \mathbf{A})$. Using the properties of the scalar triple product, we know:

$\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \mathbf{B} \cdot (\mathbf{C} \times \mathbf{A})$

Thus:

$\lambda = \mathbf{B} \cdot (\mathbf{C} \times \mathbf{A})$

So, the expression $\mathbf{B} (\mathbf{C} \times \mathbf{A})$ is equivalent to $\lambda (\mathbf{B} \times \mathbf{C})$, as both represent the scalar triple product.

Step 4: Prove the third identity

Similarly, for the third term $\mathbf{C} (\mathbf{A} \times \mathbf{B})$, we use the cyclic property of the scalar triple product:

$\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C}) = \mathbf{C} \cdot (\mathbf{A} \times \mathbf{B})$

Thus:

$\lambda = \mathbf{C} \cdot (\mathbf{A} \times \mathbf{B})$

This shows that $\mathbf{C} (\mathbf{A} \times \mathbf{B})$ is also equivalent to the other two terms.

Conclusion

We have demonstrated that all three expressions:

$\lambda (\mathbf{B} \times \mathbf{C}), \quad \mathbf{B} (\mathbf{C} \times \mathbf{A}), \quad \mathbf{C} (\mathbf{A} \times \mathbf{B})$

are equivalent, since they all represent the scalar triple product $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$. Therefore, the given identity is proven.

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Would you like more details on any part of this proof? Let me know!

Here are 5 related questions:

1. What is the geometric interpretation of the scalar triple product?
2. How do vector identities apply in physics, particularly in torque and angular momentum?
3. Can the scalar triple product ever be negative? What does that represent geometrically?
4. How does the scalar triple product relate to determinants of matrices?
5. What is the relationship between the cross product and the area of a parallelogram?

Tip: The scalar triple product $\mathbf{A} \cdot (\mathbf{B} \times \mathbf{C})$ is zero when the vectors are coplanar. This is a useful condition in solving geometric and physical problems!