# What is the cross product of [3,2, 5] and [2, -5, 8] ?

Mar 12, 2018

By hand and then checked with MATLAB: [41 -14 -19]

#### Explanation:

When you take a cross product, I feel like it makes things easier to add in the unit vector directions $\left[\hat{i} \hat{j} \hat{k}\right]$ which are in the x, y, and z directions respectively.

We'll use all three since these are 3-D vectors that we're dealing with. If it was 2d you'd only have to use $\hat{i}$ and $\hat{j}$

Now we set up a 3x3 matrix as follows (Socratic doesn't give me a good way to do multidimensional matrices, sorry!):

$| \hat{i} \hat{j} \hat{k} |$
$| 3 2 5 |$
$| 2 - 5 8 |$

Now, starting at each unit vector, go diagonal from left to right, taking the product of those numbers:

$\left(2 \cdot 8\right) \hat{i} \left(5 \cdot 2\right) \hat{j} \left(3 \cdot - 5\right) \hat{k}$

$= 16 \hat{i} 10 \hat{j} - 15 \hat{k}$

Next, take the products of the values going from right to left; again, starting at the unit vector:

$\left(5 \cdot - 5\right) \hat{i} \left(3 \cdot 8\right) \hat{j} \left(2 \cdot 2\right) \hat{k}$

$= - 25 \hat{i} 24 \hat{j} 4 \hat{k}$

Finally, take the first set and subtract the second set from it

$\left[16 \hat{i} 10 \hat{j} - 15 \hat{k}\right] - \left[- 25 \hat{i} 24 \hat{j} 4 \hat{k}\right]$
$= \left(16 - \left(- 25\right)\right) \hat{i} \left(10 - 24\right) \hat{j} \left(- 15 - 4\right) \hat{k}$
$= 41 \hat{i} - 14 \hat{j} - 19 \hat{k}$

this can now be re-written in matrix form, with $\hat{i}$, $\hat{j}$, and $\hat{k}$ removed since it's staying a 3-D vector:

$\textcolor{red}{\text{[41 -14 -19]}}$