My textbook: '.. the length of a vector is in many ways analogous to the absolute value of a real number.'
My question: How are the length of a vector and the absolute value of a real number 'analogous in many ways' and not simply equivalent? In what ways are they different?
Similar algebraic relationships hold: $|kx| = |k||x|$ where $k,x$ are numbers or $k$ is number and $x$ is a vector (these are analogous but not exactly the same, i.e. not equiv). Also consider the triangle inequality: $|x+y| \leq |x| + |y|$ where $x,y$ are number or $x$ and $y$ are vectors.
Actually, the real line is a special kind of vector space, and absolute value measures vector length in that space. The only difference is that vector length is a more general notion: it applies also to vectors in higher dimensions.
(But generally, they are analogous in the sense that they both tell you the magnitude of the entity, but ignore its `direction': absolute value of a real number ignores the sign (whether it is pointing up or down, is it were). In the more general setting the length of a vector merely tells you its length but not the direction.)
