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# Matrix multiplication and vector cross product Watch

1. If you have two vectors a and b, I was wondering why you can't just write these vectors as column vectors and treat them as matrices. Why when you multiply the two matrices is it not the same as the vector cross product? (And how do you know which one to use...)

(I've done some googling and know there's a formula to convert between the two but was just wondering if someone could explain mathematically why it is necessary - self teaching FP3 and the book doesn't explain things very well!)

Thanks
2. (Original post by Conjecture)
If you have two vectors a and b, I was wondering why you can't just write these vectors as column vectors and treat them as matrices. Why when you multiply the two matrices is it not the same as the vector cross product? (And how do you know which one to use...)

(I've done some googling and know there's a formula to convert between the two but was just wondering if someone could explain mathematically why it is necessary - self teaching FP3 and the book doesn't explain things very well!)

Thanks
The vector cross product will always give a vector of the same dimension as the two vectors that you "crossed"

However, note that you can't multiply two matrices unless the number of columns in the first one is the same as the number of rows in the second.

Does that make sense?

3. (Original post by Conjecture)
If you have two vectors a and b, I was wondering why you can't just write these vectors as column vectors and treat them as matrices. Why when you multiply the two matrices is it not the same as the vector cross product? (And how do you know which one to use...)

(I've done some googling and know there's a formula to convert between the two but was just wondering if someone could explain mathematically why it is necessary - self teaching FP3 and the book doesn't explain things very well!)

Thanks
Not entirely sure what you're asking, but typically a vector is used to represent the position of a point (or to describe a direction from the origin), whereas a matrix represents a transformation such as a rotation or reflection.

You can write vectors adjacent to each other in the form of a matrix e.g. to represent the vertices of a plane figure such as a sqaure, and then apply a transformation matrix to this matrix of coordinates - the output then gives you the image of the points under the transformation.

The vector cross product takes 2 vectors as input and produces a third vector orthogonal to the other two.

Matrix multiplication represents the composition of 2 (or more) transformations, so is not the same thing!
4. The simplest explanation is that although these two operations both have the word "multiplication" in them, they are completely different things. It can be easily proven that the two operations return different values, but are you instead asking why "matrix multiplication" is a different operation than "cross product"? If so, then I think only the person who named each operation can tell you the answer to that ^^
5. (Original post by davros)
Not entirely sure what you're asking, but typically a vector is used to represent the position of a point (or to describe a direction from the origin), whereas a matrix represents a transformation such as a rotation or reflection.

You can write vectors adjacent to each other in the form of a matrix e.g. to represent the vertices of a plane figure such as a sqaure, and then apply a transformation matrix to this matrix of coordinates - the output then gives you the image of the points under the transformation.

The vector cross product takes 2 vectors as input and produces a third vector orthogonal to the other two.

Matrix multiplication represents the composition of 2 (or more) transformations, so is not the same thing!
Thankyou, this makes sense I think I was just being a little thick... doing 6 modules in 4 months and everything's starting to get a bit muddled together :/
6. (Original post by Conjecture)
Thankyou, this makes sense I think I was just being a little thick... doing 6 modules in 4 months and everything's starting to get a bit muddled together :/
no problem!

The important thing to remember is that as mathematicians invent more concepts, there are only a finite number of ways of saying "combine this with this", so you often find operations called "multiply" or "product" which are doing subtly different things depending on the objects they work with.

A lot of matrix/vector work is just practice, so keep at it till it clicks and post on TSR if you get stuck on specific questions

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