Let
denote the set of all triples
with
Elements in
are called vectors. The sum of vectors
is
The scalar multiplication of
with the vector
is
The zero vector is denoted
For
is defined by
These rules have properties
for
for
for
for
for
for
for for
for for
translation:
magnification:
contraction:
parallelogram construction
The dot product of two vectors
and
is
Proposition
for
for
for
for
for
for
unless for (Proof. )
The equality holds if and only if there exist with
Proof. Case 1. Then so the equality holds.
Case 2.
Then
Consider the vector
Then
Therefore
If and then so i.e. If and then
The norm of the vector
is
It follows from Cauchy-Schwarz inequality that
if
The unique angle
such that
is called the angle between the vectors
and
This is consistent with the Law of Cosine:
for all if and only if
for
Triangle Inequality: for
Proof.
for
Parallelogram Law: for
Proof. Add the identities
Two vectors
are perpendicular
if the angle between them equals
Theorem. Let
Each
can be uniquely expressed as
where
is parallel to
(i.e., there exists
with
and
is perpendicular to
(i.e.,
Proof. Set
and
Then
If
with
then
so
;
this shows
and
Definition is called the projection of the vector on the vector
Notation:
If
then the vector
satisfies
i.e.,
is a unit vector. The components of
in the directions
are called the direction cosines of
Let
Their cross product (also called vector product) is given by
for
for
for
for
for for
Scalar Triple Product:
Proof.
Therefore
for
Hence is perpendicular to both and
Lagrange Identity: for
Proof.
is the area of the parallelogram formed with sides and
Proof. We may assume that
and
From the identity
we see that
where
is the angle between
and
It follows that
therefore the required area equals
The absolute value of the triple product represents the volume of the parallelepiped with edges
Proof. Write
The quantity
is the component of the vector
along the direction
so the height of the parallelepiped with base formed by the vectors
is
The result follows from the fact
CAB Minus BAC Identity
The result holds for
(there are altogether 27 different cases.) Write
The general case follows from
Extended Lagrange Identity:
Proof.
It follows from CAB minus BAC identity that this equals
Note that the Lagrange identity may be obtained from the extended Lagrange
identity by setting
and
Jacobi Identity:
Proof. Add the identities
Consider the simultaneous set of equations
From the basic properties of the determinant we have
Hence if
is to satisfy equations (1) then
Write
If
then
Similarly, if
then
On the other hand, the following identity always holds:
This four-by-four determinant can be expanded along the first row:
Hence if
then
is indeed the (unique) solution to the system (1).
The method can be extended to linear equations in unknowns.
A straight line in
can be given by the parametric equations:
A point
belong to this line if and only if
for some
This can therefore be written in vector equations as:
After eliminating the parameter
from the parametric equation, we obtain the symmetric equation
if
Example Find the equations of the straight line passing through and .
First we determine the direction:
from this the vector equation is obtained:
Equating the components we obtain the parametric equation:
and the symmetric equations:
Example Find the distance between the skew lines
given by
and
given by
We note that
has direction
and
has direction
The vector
is perpendicular to both
and
Choose any point
and any point
(There are infinitely many ways to do so.) The distance between
and
is the norm of the projection of
on
Take
and
Then
and so the required distance is
Let
be any point of the plane. The plane consists of all the points
satisfying the vector equation
for a nonzero vector
This vector
is called the normal of the plane. Thus the equation of the
plane takes the form
where
Example Find the distance from to the plane
The vector
is normal to the plane. Choose a point on the plane:
the distance from
to the plane is the norm of the projection of the vector
on
Example Show that the line
can be expressed as the intersection of two planes, each of which is parallel
to a coordinate axis.
Consider the plane
with equation obtained from the first equality
can be expressed by the equation
which has
as normal.
is therefore parallel to the
-axis
since
is perpendicular to
Similarly, the plane satisfying the equation
which is the same as the equation
has
as normal, must be parallel to the
-axis
since
is perpendicular to
The given line is the intersection of
and
The angle between two intersecting planes is the same as the
angle between their normals
.
Depending on the choices of the normals, there are two such angles, each the
supplement of the other. We choose the smaller angle, the one with the
nonnegative cosine:
Example Find the cosine of the angle between the planes with
equations
and
Solution:
;
so
Example The planes
and
intersect to form a line. Find the vector equation of this line.
Solution: The direction vector of the intersecting line must be perpendicular
to both normal vectors
and
Hence it is parallel to
Let
be any point on the intersecting line. The required vector equation takes the
form
Suppose that the points
do not lie on a straight line. A point
lies on the plane determined by these three points if and only if the volume
of the parallelepiped with
as vertices is zero. In view of the vector triple product identity, the
condition is given by
Example Find the equation for the plane passing through
Solution: Expanding the determinant
we have
Suppose that
does not lie on the plane
Choose any point
on the plane. The distance from
to the plane is the norm of the projection of
on the normal
of the plane:
Since
lies on the plane, it follows that
Therefore
consequently the required distance is given by
Find the equation for the plane
through perpendicular to
through perpendicular to
through parallel to
through parallel to
Explain why a plane cannot
contain and and be perpendicular to
contain and
go through the origin and have the equation
Find the projection of along and also find
unit vector at angle with
vector of length at angle with
is perpendicular to
is perpendicular to
True or false:
never equals
If and then either or
If and then
Which of the following equals to