# What is the cross product of <8, 4 ,-2 > and <-1, 3 ,2 >?

Jan 8, 2016

$< 14 , - 14 , 28 >$

#### Explanation:

Cross multiplication of 2 vectors $\left(a , b , c\right)$ & $\left(d , e , f\right)$ is given by:

$\left(\begin{matrix}i & j & k \\ a & b & c \\ d & e & f\end{matrix}\right)$

where $\left(i , j , k\right)$ is the unit vector in the resulting vector direction i.e. perpendicular to the plane of the two vectors being cross multiplied
So, you solve the matrix by cross multiplication and get the coefficients of $\left(i , j , k\right)$ as

$i \left(b f - c e\right) - j \left(a f - c d\right) + k \left(a e - b d\right)$

this is the resulting vector of the cross multiplication of 2 vectors $\left(a , b , c\right)$ & $\left(d , e , f\right)$

In our case, the vectors are $\left(8 , 4 , - 2\right)$ & $\left(- 1 , 3 , 2\right)$
So,

$\left(\begin{matrix}i & j & k \\ 8 & 4 & - 2 \\ - 1 & 3 & 2\end{matrix}\right)$

$i \left[4 \left(2\right) - 3 \left(- 2\right)\right] - j \left[8 \left(2\right) - \left(- 1\right) \left(- 2\right)\right] + k \left[8 \left(3\right) - \left(- 1\right) \left(4\right)\right]$

$i \left(8 + 6\right) - j \left(16 - 2\right) + k \left(24 + 4\right)$

= $14 i - 14 j + 28 k$ or $< 14 , - 14 , 28 >$