I am asked this: How can you do the cross product of two-dimensional vectors. Well I would have to guess that you just add a 0 in order to follow the cross product method, but this way there is no third component...is this right?
(ai+bj)X (ci+dj)=ad k + bc (-k)
so the cross product is only in the third direction.
I don't really understand what you mean, sorry. Can you explain it differently?
given the three dimensional unit vectors, i,j,and k, the cross product of ai+bj with ci+dj is as I calculated. See
http://en.wikipedia.org/wiki/Cross_product
The Cross Product gives us the normal vector to the plane that both vector a and vector b lie on.
Therefore, the cross product will always result in 3 dimensions
The cross product is a mathematical operation defined for 3-dimensional vectors. So, it cannot be directly applied to 2-dimensional vectors. However, there is a way to extend the cross product to 2-dimensional vectors by considering them as 3-dimensional vectors with a fixed z-component of zero.
To calculate the cross product of 2-dimensional vectors A = (Ax, Ay) and B = (Bx, By), you can follow these steps:
1. Extend the vectors to 3 dimensions: Append a zero to the z-component of both vectors. So A becomes (Ax, Ay, 0) and B becomes (Bx, By, 0).
2. Perform the cross product as you would for 3-dimensional vectors: The cross product of two 3-dimensional vectors is given by the formula:
A × B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx).
Since the z-component for both A and B is zero, the resulting cross product will have no contribution for the z-component.
3. Simplify the resulting cross product: Since the z-component is zero, you can ignore it, and the resulting cross product will be a 2-dimensional vector.
Therefore, the cross product of the 2-dimensional vectors A and B would simply be:
A × B = (Ay*0 - 0*By, 0*Bx - Ax*0) = (0, 0).
As you can see, the resulting cross product of two 2-dimensional vectors will always be the zero vector since there is no third component.
So, to answer your question, yes, you would add a zero in order to apply the cross product method to two-dimensional vectors, but the result will always be a 2-dimensional zero vector.