Confusion regarding a number in a cross product problem.

Question

I'm practicing cross product and everything is going fine until I reach a certain problem. The vectors I'm given are <5, -3, 3> and <4,-6, 0> respectively. My answer is <18,-12, 42>. However, the website I'm using says the answer is <18, 12, 42>. I'm stumped on how they got 12, seeing as the top vector is <5, -3, 3> and the bottom vector is <4, -6, 0>. The only way to get 12 is to do (4 * 3) - (-5 * 0), but isn't that breaking a rule? A little clarification would be nice. Here's a picture. if it helps any.

Answer 1

Here's a technique you can use:

Write an array of numbers made of the components of your vectors. The first vector goes on top and the second vector goes on the bottom. Ignore the first and last columns. \begin{array}{} \color{gray}{5} & -3 & 3 & 5 & -3 & \color{gray}{3}\\ \color{gray}{4} & -6 & 0 & 4 & -6 & \color{gray}{0} \end{array}

The $x$ component of your vector product is the difference in blue and red products: $$\begin{array}{} \color{gray}{5} & \color{blue}{-3} & \color{red}{3} & 5 & -3 & \color{gray}{3}\\ \color{gray}{4} & \color{red}{-6} & \color{blue}{0} & 4 & -6 & \color{gray}{0} \end{array}\implies\color{blue}{0}-(\color{red}{-18})=18.$$

Then, move over to find the $y$ component. $$\begin{array}{} \color{gray}{5} & -3 & \color{blue}{3} & \color{red}{5} & -3 & \color{gray}{3}\\ \color{gray}{4} & -6 & \color{red}{0} & \color{blue}{4} & -6 & \color{gray}{0} \end{array}\implies\color{blue}{12}-\color{red}{0}=12.$$

and once more for the $z$ component. $$\begin{array}{} \color{gray}{5} & -3 & 3 & \color{blue}{5} & \color{red}{-3} & \color{gray}{3}\\ \color{gray}{4} & -6 & 0 & \color{red}{4} & \color{blue}{-6} & \color{gray}{0} \end{array}\implies\color{blue}{-30}-(\color{red}{-12})=-18.$$

Thus, the vector product is $(18,12,-18)$.