| (2, -4, 4)T | = √( 2 * 2 + -4 * -4 + 4 * 4) = √( 4+ 16 + 16 ) = √36 = 6
| (-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 (or zero) 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 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 4-component column matrices to represent vectors in three dimensions. Calculating the length of a vector represented in that manner calls for a modification of the formulas (the fourth component of the column is ignored.)
Don't worry about that now. Details will come soon enough. But do take the time to become comfortable about the idea that the column matrices 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.
Is the distinction between an object and its representation of any importance in computer science?