The cross product of two vectors $\langle a_1, a_2, a_3 \rangle$ and $\langle b_1, b_2, b_3 \rangle$ is given by:

$\langle a_1, a_2, a_3 \rangle \times \langle b_1, b_2, b_3 \rangle = \langle a_2b_3 - a_3b_2, a_3b_1 - a_1b_3, a_1b_2 - a_2b_1 \rangle$

Given the vectors $\langle 5, 1, 2 \rangle$ and $\langle -2, 1, -3 \rangle$, let's compute each component:

1. First Component: $a_2b_3 - a_3b_2 = (1)(-3) - (2)(1) = -3 - 2 = -5$
2. Second Component: $a_3b_1 - a_1b_3 = (2)(-2) - (5)(-3) = -4 + 15 = 11$
3. Third Component: $a_1b_2 - a_2b_1 = (5)(1) - (1)(-2) = 5 + 2 = 7$

Thus, the cross product $\langle 5, 1, 2 \rangle \times \langle -2, 1, -3 \rangle$ is:

$\langle -5, 11, 7 \rangle$

Would you like further details or explanations?

Here are 5 related questions for further exploration:

1. What is the magnitude of the cross product vector $\langle -5, 11, 7 \rangle$?
2. How do you verify if two vectors are parallel using their cross product?
3. What is the geometric significance of the cross product?
4. How can you determine the area of the parallelogram formed by two vectors using their cross product?
5. What is the relationship between the cross product and the sine of the angle between two vectors?

Tip: The direction of the cross product follows the right-hand rule, which helps determine the orientation of the resulting vector.