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Geometry
Vectors, points and arrows
A 3×1 column-vector or 3-vector (x y z)T
determines a point (x, y, z) in the three-dimensional
Cartesian space. Conversely, every point in this space is determined by a vector.
For any point (a, b, c) in space, the vector (x y
z)T also determines an arrow (or directed
line segment, if you like) with tail at that point and
head at the point
(a + x, b + y, c + z). (The
book refers to such an arrow as a vector, but I shall avoid doing so.)
One purpose of allowing the tail-point of an arrow to vary is this.
Suppose x is the vector (x1 x2
x3)T, and y is the vector
(y1 y2 y3)T.
Then x determines the arrow from the origin (0, 0, 0) to
(x1, x2, x3),
and y determines the arrow from this point to
(x1 + y1, x2 +
y2, x3 + y3),
and this is the point determined by x + y.
Norm
The norm of a 3-vector is the length of the arrow from the
origin to the point that the vector determines. If u is the
vector (u1 u2
u3)T, then the norm of
u is the non-negative
scalar |u| such that
|u|2 = u12 +
u22 + u32
.
(Note that the norm is often indicated by double bars, rather than
single as here.) In particular, if k is a scalar, then
|ku| = |k||u| ,
that is, the norm of ku is the product of the absolute
value of k and the norm of u.
Dot-product
Definition
Suppose also v is the vector (v1 v2
v3)T. Then each of u and
v determines an arrow with tail-point at the origin; let
0 (Greek small letter theta) be the angle from the
one arrow to the other (assuming the arrows have positive length).
With the origin, the head-points determine a triangle; we have
|u - v|2 =
|u|2 + |v|2 -
2|u||v| cos 0
by the Law of Cosines. The dot-product
of u and v is the scalar
|u||v| cos 0
appearing in the equation; it is denoted u·v.
The Law of Cosines yields a formula:
u·v = u1v1 +
u2v2 + u3v3
;
this can be taken as an alternative definition of the dot-product, and
it makes sense regardless of the norms of the vectors.
If the norms of u and v are positive, then angle
0 is acute, right or obtuse, respectively, if and
only if u·v is positive, zero or negative. Regardless of the
norms, we shall say that two vectors are orthogonal if their
dot-product is zero. [The symbol for orthogonality is an upside-down T.]
Properties
The dot-product is an example of a scalar product of vectors; we
shall see others later. The dot-product is like a product of
scalars in the following ways:
- It is symmetric: u·v = v·u.
- It distributes over addition: u·(v + w) =
u·v + u·w.
- It associates with scalar multiplication: k(u·v) =
ku·v.
- v·v > 0 always, and
v·v = 0 if and only if v = 0.
The last property holds because
|u|2 = u·u.
Projections
Suppose a is a non-zero vector. There is a vector
ka such that u - ka is orthogonal to
a (and hence to ka). In fact, we can calculate that
k = (u·a)/(a·a).
We call ka the projection of u
on a and denote it by projau. We may
say that projau is parallel to
a, since it is a scalar multiple of a. So then u
is the sum of two orthogonal vectors, namely
projau and u -
projau, the first being parallel to
a.
Cross-product
The appropriate changes being made, everything till now in this
section makes sense in the Cartesian plane. The same is not true for
the cross-product, which is defined only in 3-space.
Theoretical definition
If u and v are 3-vectors, then there is a 3-vector
w such that
det (u v x) = w·x
for any 3-vector x. The vector w is called the
cross-product of u and v, denoted
u×v.
Practical definition
You can check that the previous definition justifies the following method for
calculating the cross-product. Suppose i, j and
k are the vectors determining the points (1, 0, 0), (0, 1, 0)
and (0, 0, 1) respectively. You can write down what looks like a 3×3
matrix, with u as the first column, and v as the second
column, but where the entries of the third column are i,
j and k respectively. If you calculate the determinant
formally, you get a linear combination of i, j and
k, which is a vector itself: it is the cross-product
u×v.
Properties
The cross-product is anti-symmetric: v×u =
-u×v.
From the theoretical definition, it follows that
u×v is orthogonal to both u and v.
Using the facts about dot-products, you can also show
|u×v| = |u||v|sin
0.
This means that the norm of the cross-product of u and v
is the area of the parallelogram determined by u and
v (that is, the parallelogram whose vertices are
the origin and the points determined by u, v and
u + v).
If there is a third 3-vector, x, then the three vectors
u, v and x determine a parallelepiped.
[The word is Greek: parallel + epi +
ped[on]; an epipedon is a surface; compare
parallel + o + gram[mon], where gramma is
a line.] The next equation is a restatement of the theoretical
definition of the cross-product:
(u×v)·x = det (u
v x).
The volume of the parallelepiped is the absolute value of either side of this
equation.
In 2-space, two vectors determine a parallelogram, whose area is the
absolute value of the determinant of the matrix whose columns are the
two vectors.
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