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The Cross Product


The cross product of a and b, written a x b, is defined by:
a x b = n a b sin q
where a and b are the magnitude of vectors a and b; q is the angle between the vectors, and n is the unit vector (vector with magnitude = 1) that is perpendicular (at 90 degrees to/ orthogonal to/ normal to) both a and b. But there are two vectors that this could be - one on either side of the plane formed by the two vectors), so we choose n to be the one which makes (a, b, n) a
right handed triad.

Like in the definition of the dot product where we pulled q out of a hat and said it was the angle between the two vectors without any way of finding it, so we need a way of finding n for out definition of the cross product to be any use. Again the i, j, k vectors come to our rescue, giving us an equivalent definition: let,
a = a1 i + a2 j + a3 k
b = b1 i + b2 j + b3 k
a x b = ( a1 i + a2 j + a3 k) x (b1 i + b2 j + b3 k)
The cross product of any two parallel vectors is the null vector since sin 0 = 0, and also
i x j = k
j x k = i
k x i = j
j x i = -k
k x j = -i
i x k = -j
Using these, we can eventually find:
a x b = (a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k
That's our equivalent definition. If you're familiar with determinants you may see this can be written more conveniently as,

    | i    j    k|        
    | a1  a2   a3|  
    | b1  b2   b3| 

Finding the normal vectors

Using the equivalence of our two definitions,
a x b = (a2b3 - a3b2)i + (a3b1 - a1b3)j + (a1b2 - a2b1)k = n a b sin q
you can now find n
To get used to the cross product try out
this cross product applet

Some Properties of the Cross Product

The cross product is anticommutative:
a x b = - b x a
The cross product of parallel vectors is the null vector, in particular:
a x a = 0
Also | a x b | is the area of the parallelogram formed by a and b,

Area = a b sin q

Right handed triads

Here (x, y, z) (y, z, x) (z, x, y) form a right handed triad while (x, z, y) (z, y, x) (y, x, z) form a left handed triad

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