# Cross Product (Algebraic)

Vector multiplication can be completed using the dot product or cross product. The result of the cross product is a vector. We can calculate the cross product using geometric or algebraic vectors. The algebraic cross product of two vectors $$\vec{a} = \left(a_{1},a_{2},a_{3}\right), \vec{b} = \left(b_{1},b_{2},b_{3}\right)$$ is defined by:

$$\vec{a}\times\vec{b} = \left(a_{2}b_{3}-a_{3}b_{2},a_{3}b_{1}-a_{1}b_{3}, a_{1}b_{2}-a_{2}b_{1} \right)$$

Given two vectors $$\vec{a} = (2,4,6)$$ and $$\vec{b} = (-1,2,-5)$$ calculate $$\vec{a}\times\vec{b}$$ and $$\vec{b}\times\vec{a}$$. What property does this demonstrate does hold not for the cross product? Explain why the property does not hold.

Using the two vectors above and a third vector $$\vec{c} = (4,3,-1)$$ calculate
1) $$\vec{a}\times(\vec{b}+\vec{c})$$ and 2) $$(\vec{a}\times\vec{b}) + (\vec{a}\times\vec{c})$$

Problem 1) $$\vec{a}\times(\vec{b}+\vec{c})$$

Problem 2) $$(\vec{a}\times\vec{b}) + (\vec{a}\times\vec{c})$$

Compare the results from above. What property does this demonstrate?

Using the three vectors $$\vec{a}, \vec{b} \text{ and } \vec{c}$$ from above calculate
1) $$(\vec{a}\times\vec{b})\times\vec{c}$$ and 2) $$\vec{a}\times(\vec{b}\times\vec{c})$$

Problem 1) $$(\vec{a}\times\vec{b})\times\vec{c}$$

Problem 2) $$\vec{a}\times(\vec{b}\times\vec{c})$$

What property does this demonstrate does NOT hold for the cross product?