The given problem involves vectors $a = \langle 4, -2, 5 \rangle$, $b = \langle -1, 3, -6 \rangle$, and $c = \langle 7, -5, 1 \rangle$. We need to compute the cross products:

1. $a \times b$
2. $b \times c$
3. $a \times c$

I'll proceed with the calculations for each of these cross products.

Cross product formula:

For two vectors $\mathbf{u} = \langle u_1, u_2, u_3 \rangle$ and $\mathbf{v} = \langle v_1, v_2, v_3 \rangle$, the cross product $\mathbf{u} \times \mathbf{v}$ is given by: $\mathbf{u} \times \mathbf{v} = \langle u_2 v_3 - u_3 v_2, u_3 v_1 - u_1 v_3, u_1 v_2 - u_2 v_1 \rangle$

Let's compute each one now.The cross products of the vectors are as follows:

1. $a \times b = \langle -3, 19, 10 \rangle$
2. $b \times c = \langle -27, -41, -16 \rangle$
3. $a \times c = \langle 23, 31, -6 \rangle$

Let me know if you'd like more details or explanations on these calculations.

Here are 5 related questions you might explore:

1. What is the geometric interpretation of a cross product between two vectors?
2. How would you compute the magnitude of the cross product?
3. Can two parallel vectors have a non-zero cross product?
4. How do the directions of $a \times b$ and $b \times a$ compare?
5. What is the relation between the cross product and the area of the parallelogram formed by two vectors?

Tip: The cross product is anti-commutative, meaning $\mathbf{u} \times \mathbf{v} = -(\mathbf{v} \times \mathbf{u})$.