Let v = (3, 4)T
v · v = ( 3, 4 )T · ( 3, 4 )T = 32 + 42 = 9 + 16 = 25 = 52
The length of v = 5 since it is part of a 3/4/5 right triangle.
As you have seen in the previous chapter:
(x, y, z)T · (x, y, z)T = x2 + y2 + z2
Another way of writing this is:
v · v = | v |2
The dot product of a column matrix with itself is a scalar, the square of the length of the vector it represents.
WARNING! When your graphics text starts using homogeneous coordinates this calculation will need to be modified somewhat. Remember, length is a property of the geometric vector, not an inherent property of the column matrix that might be used to represent it.
What is the length of the vector represented by ( 2, 1, -1)T ?