These are notes from Cross product introduction | Vectors and spaces | Linear Algebra | Khan Academy by Khan Academy.
The Cross Product is much more limited than Dot Product. Where the dot product is defined in any dimension ( ), the cross product is only defined in 3d ( ).
The dot product returns a scalar; the cross product a Vector.
Definition of the dot product:
- For first row in the returned vector, you ignore the top row and take
- For the 2nd row in the returned vector, you ignore the middle row of the vectors and take a similar product to the first; however, this time, you are doing it the opposite way around:
- For the 3rd row, you ignore the last row of input and make the same operation as the first row of the top 2 rows of input: .
The vector that's returned is orthogonal to both and .
Note that two vectors are orthogonal to those vectors. To find which direction it points in, you use the right-hand rule: take your right hand and put your index finger in the direction of and your middle finger in the direction of , where your thumb is pointing in the direction of the returned vector.
What does orthogonal mean in this context? It means if , the difference between orthogonal vectors and perpendicular vectors is orthogonal could also apply to 0 vectors.
You can prove it works by taking the dot product with one of the input vectors and the output vector: