### Video Transcript

Calculate the cross product of π²
equals two π’ plus four π£ plus eight π€ and π³ equals six π’ plus four π£ plus two
π€.

In this example, we want to solve
for π² cross π³, where π² and π³ are both three-dimensional vectors. We can begin our solution by
recalling the mathematical definition for a cross product. The cross product of two vectors,
we can call them π and π, is equal to the determinant of a three-by-three matrix ,
where the columns are headed by the three unit vectors π’, π£, and π€. And the last two rows are populated
by the respective components of vectors π and π. We can apply this formula to our
vectors π² and π³. As we set up our matrix, we know
the top row will be the π’, π£, and π€ unit vectors. The next row will be the respective
components of the vector π². Looking at π², we see that itβs
π₯-component is two, its π¦-component is four, and its π§-component is eight. So we write those values in to our
matrix. In the next row, weβll write the
components of π³. The π₯-component of π³ is six. Its π¦-component is four. And its π§-component is two.

Now weβre ready to calculate the
determinant of this matrix and solve for the cross product of π² cross π³. When we compute this determinant,
we find that the π’-component is eight minus 32, or negative 24. The π£-component is four minus 48,
or negative 44. And the π€-component is eight minus
24, which equals negative 16. Our overall cross product then is
negative 24π’ minus 44π£ minus 16π€. Thatβs the cross product of the
vectors π² and π³.