Everything you didn't want to know about dot and cross product

Everything you didnt want to know about dot and cross product

The dot product is the product of the cosine of the angle between two vectors and their length/magnitude. It can be written it as:

   A⋅B = |A||B|cos theta

The dot product is commutative:
```
   A⋅B = B⋅A
```
also for any scalar k:
```
   (kA)⋅B = k(A⋅B)
```
It should also be noted that the operation is distributive
```
   A⋅(B+C) = A⋅B + A⋅C
```

Now define 3 perpendicular vectors i, j and k, we can now state:

i.i = j.j = k.k = 1
i.j = j.k = k.i = 0

If we have our vectors A and B in component form:

A = i*x1 + j*y1 + k*z1
B = i*x2 + j*y2 + k*z2

We can then find a component form expression for the dot product.

A⋅B = (i*x1 + j*y1 + k*z1)⋅(i*x2 + j*y2 + k*z2)

    = (i*x1 + j*y1 + k*z1)⋅(i*x2)
    + (i*x1 + j*y1 + k*z1)⋅(j*y2)
    + (i*x1 + j*y1 + k*z1)⋅(k*z2)

    = (i*x1)⋅(i*x2) + (j*y1)⋅(i*x2) + (k*z1)⋅(i*x2)
    + (i*x1)⋅(j*y2) + (j*y1)⋅(j*y2) + (k*z1)⋅(j*y2)
    + (i*x1)⋅(k*z2) + (j*y1)⋅(k*z2) + (k*z1)⋅(k*z2)

    = (i⋅i)*x1*x2 + (j⋅i)*y1*x2 + (k⋅i)*z1*x2
    + (i⋅j)*x1*y2 + (j⋅j)*y1*y2 + (k⋅j)*z1*y2
    + (i⋅k)*x1*z2 + (j⋅k)*y1*z2 + (k⋅k)*z1*z2

    = (1)*x1*x2 + (0)*y1*x2 + (0)*z1*x2
    + (0)*x1*y2 + (1)*y1*y2 + (0)*z1*y2
    + (0)*x1*z2 + (0)*y1*z2 + (1)*z1*z2

    = x1*x2 + y1*y2 + z1*z2

Now the cross product is defined as the product of the sine of the angle between two vectors, their magnitude and a unit vector perpendicular to those vectors.

AxB = |A||B|sin(theta)n
    = (i*x1 + j*y1 + k*z1)×(i*x2 + j*y2 + k*z2)

We can make the following observations about the cross of the perpendicular vectors i, j, k:

x	i	j	k
i	0	k	-j
j	-k	0	i
k	j	-i	0

It should be noted this table can be made in two ways, left or right handed. So there are really two definitions of cross product but it's conventional to use this one.

The cross product is distributive like the dot product.

Again in component form:

A×B = (i*×1 + j*y1 + k*z1)×(i*×2 + j*y2 + k*z2)

    = (i*×1 + j*y1 + k*z1)×(i*×2)
    + (i*×1 + j*y1 + k*z1)×(j*y2)
    + (i*×1 + j*y1 + k*z1)×(k*z2)

    = (i*×1 + j*y1 + k*z1)×(i*×2)
    + (i*×1 + j*y1 + k*z1)×(j*y2)
    + (i*×1 + j*y1 + k*z1)×(k*z2)

    = (i*×1)×(i*×2) + (j*y1)×(i*×2) + (k*z1)×(i*×2)
    + (i*×1)×(j*y2) + (j*y1)×(j*y2) + (k*z1)×(j*y2)
    + (i*×1)×(j*z2) + (j*y1)×(k*z2) + (k*z1)×(k*z2)

    =     0*×1*×2 +    -k*y1*×2 +     j*z1*×2
    +     k*×1*y2 +     0*y1*y2 +    -i*z1*y2
    +    -j*×1*z2 +     i*y1*z2 +     0*z1*z2

    = i*(y1*z2-z1*y2) + j*(×1*y2-y1*×2) + k*(×1*y2-y1*×2)

If you had studied matrix math, you may notice that this is equal to the determinant of the following 3x3 matrix:

/x1 y1 z1\
|x2 y2 z2|
\ i  j  k/