(30, 40)^{T} = 10(3, 4)^{T}, so the length is 10*5 = 50
Or, if you want to do it the hard way:
|(30,40)|^{2} = 900 + 1600 = 2500. The length = √2500 = 50.
Recall that a unit vector is a vector of length one. Creating a unit vector in the same direction as a given vector is called normalizing the vector. Here is the formula for normalizing a vector (copied from a previous chapter):
v_{u} = v / |v| |
The formula works both for vectors and the column matrices that represent them. The length | v | can be calculated by using the column matrix dot product.
Potential Confusion: The word "normal" has several meanings. Normalizing a vector means making a unit vector in the same direction as the original. A vector that is perpendicular to a particular surface is sometimes called a "normal vector" but is not necessarily a unit vector. Be careful never to say "normal vector" when you mean "unit vector."
(Calculator Problem: ) Normalize the vector represented by ( 1.2, -4.2, 3.5 )^{T}.