What is the length of
| (-1, -2, 3)T | = ( -1 * -1 + -2 * -2 + 3 * 3) = ( 1 + 4 + 9 ) = 14 = 3.742
Squaring the elements of the vector results in a sum of all positive values, ensuring a positive (or zero) value for length.
Keep in mind that vectors are geometrical objects: a length and a direction in space. Vectors are represented with column matrices. The formulas for length that have been presented in this chapter assume that a coordinate frame is being used and that the vectors are represented with column matrices in that frame.
Your graphics text book will discuss how homogeneous coordinates
are used to represent vectors.
That method uses
Don't worry terribly about that now. Details will come soon enough. But do take the time to become comfortable about the idea that column matrices such as we have been using are not the only way to represent vectors, and that length is a property of the vector, not of the column matrix that represents it.