Linear Algebra

Section II. Linear Geometry of n-Space 33
An object comprised of a magnitude and a direction is a vector (we will use
the same word as in the previous section because we shall show below how to
describe such an object with a column vector). We can draw a vector as having
some length, and pointing somewhere.
There is a subtlety here — these
are equal, even though they start in different places, because they have equal
lengths and equal directions. Again: those vectors are not just alike, they are
equal.
How can things that are in different places be equal? Think of a vector as
representing a displacement (‘vector’ is Latin for “carrier” or “traveler”). These
squares undergo the same displacement, despite that those displacements start
in different places.
Sometimes, to emphasize this property vectors have of not being anchored, they
are referred to as free vectors.
These two, as free vectors, are equal;
we can think of each as a displacement of one over and two up. More generally,
two vectors in the plane are the same if and only if they have the same change
in first components and the same change in second components: the vector
extending from (a1,a2) to(b1,b2) equals the vector from (c1,c2) to(d1,d2) if
and only if b1 − a1 = d1 − c1 and b2 − a2 = d2 − c2.
An expression like ‘the vector that, were it to start at (a1,a2), would stretch
to (b1,b2)’ is awkward. Instead of that terminology, from among all of these
we single out the one starting at the origin as being in canonical (or natural)
position and we describe a vector by stating its endpoint when it is in canonical