LESSON 1
Vectors in the Plane and in Space
READ:
Sections 13.1, 13.2
NOTES:
The notion of a 3dimensional coordinate system is most likely familiar to you, and you’ve probably met
vectors before, perhaps in a physics course. The main purpose of these two sections is to quickly review
those ideas, and to establish the notation we’ll be using throughout the rest of the book.
The distance between two points in 3space is given by the length of the diagonal of the rectangular box having
those points as opposite corners, as shown in ﬁgure 4, page 667 of the text. That diagonal is the hypotenuse of
a right triangle. One side of the triangle is an edge of the box, and the other side is the diagonal of one of the
faces of the box. The diagonal of the face is itself the hypotenuse of a right triangle. First use the Pythagorean
Theorem to compute the length of the face diagonal. Then use the Pythagorean Theorem again to calculate
the length of the diagonal of the box. The result is that if
P
is the point (
x
1
,y
1
,z
1
) and
Q
is the point
(
x
2
,y
2
,z
2
) then the distance between
P
and
Q
is given by

PQ

=
p
(
x
2

x
1
)
2
+ (
y
2

y
1
)
2
+ (
z
2

z
1
)
2
.
A sphere is made up of all the points at a ﬁxed distance (called the
radius
of the sphere) from a certain ﬁxed
point (that point is called the
center
of the sphere of course). The distance formula derived above can be used
to write down an equation for the sphere with radius
r
and center at the point
C
= (
a,b,c
). In fact, the point
P
= (
x,y,z
) will be on the sphere if and only if

CP

=
r
, and according to the distance formula above, that
is the same as
p
(
x

a
)
2
+ (
y

b
)
2
+ (
z

c
)
2
=
r
, or a little more neatly, (
x

a
)
2
+(
y

b
)
2
+(
z

c
)
2
=
r
2
.
That certainly looks a lot like the equation of the 2space analog of the sphere (that is, the circle).
Vectors
are used in physics to represent a quantity that has both a magnitude and a direction associated
with it (in other words, a quantity that can be meaningfully represented by an arrow). Examples of vector
quantities are forces (10 lbs directed downwards, for example) and velocity (3 feet per second, heading
northwest). A visual representation of a vector can be provided by an arrow joining two points, say
P
and
Q
, in space. The magnitude of the vector is represented by the distance between the points, and the direction
of the vector is indicated by the straight line from the ﬁrst point,
P
, to the second point,
Q
.
In writing the vector from
P
to
Q
, we’ll use the notation
→
PQ
, and we’ll draw an arrow from
P
(the
initial
point of the vector) to
Q
(the
terminal
point of the vector). Two vectors are called
equal
if they have the
same magnitude and the same direction. The particular initial and terminal points are not important as far
as the vector is concerned. If two arrows have the same length and are parallel (and pointing in the same