Expert: David Montiel - 8/16/2011

Question
Hello David, Here is my question and below it I will show you what I have done:

Q: For the following pair of vectors, find the cross product using distributivity.

A = (0,2,1) B = (1,2,0)

I know how to find the cross product by using determinates, which would give me the answer of:

-2x + y -2z (where x,y and z are unit vectors).

For distributivity I would write it as:

A x B = (0x + 2y + z)x(x + 2y + oz)
A x B =(2y + z) x (x + 2y)

I'm not sure to do after this to arrive at the same answer for when I use the determinant method.

Thank you in advance.

Answer
Hello Jay

First of all lets denote the "cross" sign as "*" to avoid confusion with the unit vector "x". For example

A * B = A [cross] B.

OK?

Now, with the distributivity method you were on the right track. All you need to do is carry out the product of each pair as you would do with algebraic expressions like, for example:

(a+b)(c+d) = ac+ad+bc+bd.

In your case, every term will be a number times a cross product between two unit vectors. Something like

(2y+x) * (x+2y) = 2 y*x +...

So you in the end you will have four terms and on each term there will be a product of unit vectors. You should know (or use the right hand rule to figure out) what each pair of unitary vectors give when you cross them. For instance

x*y=z.

Once you've carried out the cross product on each term you should simplify your answer so that it looks look like the one you got using determinants.

Hope this helps!

David

ps- Be very careful when you (cross) multiply each term because the cross product is not commutative, so make sure you multiply in the right order. For instance

x*z is not equal to z*x