Mathematical Structure -- Vector Spaces
Magnitude or Length
Prerequisites:
This module follows a pattern that will become quite familiar. We begin
by looking at a geometric idea in R2 where it is easy to draw
pictures. Then we look at the same idea in R3 where the pictures
are a little more difficult but we still have lots of geometric and
physical experience. Then we move on to more exotic vector spaces
where the geometry is no longer so evident but the algebraic machinery that
we have developed still works. This allows us to use geometric ideas born
in R2 and R3 to solve problems in more exotic settings.
If we think of a vector v = (x, y) in R2 as an arrow
starting at the origin as shown in the figure below
then the Pythagorean Theorem tells us that the length of this arrow is
We use the notation ||v|| for the length or
magnitude of a vector v.
Similarly, if we think of a vector v = (x, y, z) in R3
as an arrow starting at the origin as shown in the figure below
then once again the Pythagorean Theorem allows us to compute its length. First
we consider the vector
L = (x, y, 0).
This vector is the
hypotenuse of a right triangle whose three vertices are at the points
(0, 0, 0) , (x, 0, 0) , and (x, y, 0),
so its length is
Next consider the right triangle whose vertices are at the points
(0, 0, 0), (0, 0, z), and (x, y, z).
The vector (x, y, z)
is the hypotenuse of this right triangle and its other two sides have
length ||L|| and |z|, so its length is
We can generalize the formula above to vectors in Rn
We can use this formula to compute the distance between two points, represented
by vectors u and v. The distance between two points is the
length of the movement v - u required to move from one to the other --
that is,
The distance between u and v is
||v - u||.
This notion of length or magnitude has the following
properties, called the magnitude properties.
- If v is a vector then ||v|| >= 0 and ||v|| = 0
if and only if v = O.
- If c is a real number and v is a vector then
||c v|| = |c| ||v||.
- If u and v are vectors then
||u + v|| <= ||u|| + ||v||.
This last property is called the triangle inequality. Geometrically
it says that the length of one side, u + v, of a triangle is less than
or equal to the sum of the lengths of its other two sides, u
and v.
- Find the length of the vector (3, 4).
answer
- Find the length of the vector (3, 4, 5).
answer
- Find the length of the vector (3, 4, 5, 6).
answer
- Prove the first of the magnitude properties above --
if v is a
vector then ||v|| >= 0 and ||v|| = 0
if and only if v = O.
answer
- Prove the second of the magnitude properties above --
if c is a
real number and v is a vector then
||c v|| = |c| ||v||.
answer
Note: We will not prove the triangle inequality here. It will be proved
in the module on the dot product.
- Let C[a, b] denote the set of all continuous functions on the
interval [a, b]. Define the usual two vector space operations
on this set of functions. That is, if u and v are two
functions in C[a, b] and c is a real number then the function
u + v is given by
(u + v)(t) = u(t) + v(t)
and the function c u is given by
(c u)(t) = c u(t).
Define ||u|| by
Show that this notion of magnitude satisfies the first two magnitude
properties.
answer
Copyright c 1995 by
Frank Wattenberg, Department of Mathematics, Montana State University,
Bozeman, MT 59717