Section 12.2
Vectors
In this section, we will always assume that we are in 3-dimensional space (unless otherwise
stated), so that all points are described by three coordinates.
Many quantities, such as velocity and force, need more than just information about their magniutede in order to completely describe them; in particular, to talk about either velocity or force,
we must specify the direction of the quantity. Velocity and force are examples of vectors:
Definition 0.0.1. A vector is a quantity described by a length (magnitude) and a direction.
Figure 1: Vector: Committing crime with both direction and magnitude
⃗ shown below has initial
We can think of a vector as a directed line segment. The vector AB
⃗
point A and terminal point B. The length of the vector is denoted |AB|.
1
Section 12.2
We will often need to compare two vectors to see whether or not they are equal. Since a vector
is completely described by its length and direction, then two vectors with the same length and
direction are said to be equal, even when they are located in different areas of space. The vectors
in the following diagram are all equal:
To make it clear that two vectors are indeed equal, it is often convenient to move them in space
so that they share an initial point; then the vectors are equal if they share terminal points. The
easiest way to compare two vectors is to use the origin (0, 0, 0) as initial point.
If the vector ⃗u is equal to vector ⃗v , where ⃗v has initial point (0, 0, 0), then we call ⃗v the position
vector of ⃗u. Any vector equal to ⃗u also has position vector ⃗v .
2
Section 12.2
Identifying a vector ⃗u with its position vector ⃗v gives us a convenient way to refer to vectors.
Since the position vector ⃗v always has initial point (0, 0, 0), we only need to refer to its terminal
point (v1 , v2 , v3 ) to specify ⃗v . Since ⃗u and ⃗v are equal, we can use this information to refer to ⃗u as
well.
Definition 0.0.2. If the vector ⃗u has position vector ⃗v , with the terminal point of ⃗v at (v1 , v2 , v3 ),
then the component form of ⃗u is given by
⃗u = ⟨v1 , v2 , v3 ⟩.
Notice that, although the component form of ⃗u tells us the length and direction of ⃗u, it does
not specify its location in space. Two equal vectors that occupy different locations in space have
the same component form.
To find the component form of a vector ⃗u with initial point (x1 , y1 , z1 ) and terminal point
(x2 , y2 , z2 ), we need to shift the vector so that its initial point is (0, 0, 0). We can accomplish
this by moving the x coordinate of the initial point by −x1 , the y coordinate by −y1 , and the z
coordinate by −z1 :
3
Section 12.2
We need to shift the terminal point by the same amount. So the terminal point of the position
vector of ⃗v is given by (x2 − x1 , y2 − y1 , z2 − z1 ).
So the component form of a vector ⃗u with initial point (x1 , y1 , z1 ) and terminal point (x2 , y2 , z2 )
is given by
⃗u = ⟨x2 − x1 , y2 − y1 , z2 − z1 ⟩
We would like to be able to calculate the length of a vector; since we know how to find the
distance between two points in 3-space, the problem is quite simple:
The length or magnitude of a vector ⃗u with initial point p1 = (x1 , y1 , z1 ) and terminal point
p2 = (x2 , y2 , z2 ) is just the distance from p1 to p2 , denoted
√
|⃗u| = (x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 .
Alternatively, if ⃗u has position vector ⃗u = ⟨u1 , u2 , u3 ⟩, then the length of ⃗u is
√
|⃗u| = u21 + u22 + u23 .
Example:
Find the component form and length of the vector ⃗u with initial point u1 = (3, −7, 2) and
terminal point u2 = (−4, 1, 1).
We get the component form of ⃗u by subtracting the initial point from the terminal point:
⃗u = ⟨−4 − 3, −1 − (−7), 1 − 2⟩ = ⟨−7, 8, −1⟩.
In other words, if we move ⃗u in space so that its initial point lies on the origin, then its terminal
point will be at ⟨−7, 8, −1⟩.
4
Section 12.2
To determine the length of the vector, we calculate
√
√
(x2 − x1 )2 + (y2 − y1 )2 + (z2 − z1 )2 = (−7)2 + 82 + (−1)2
√
= 49 + 64 + 1
√
= 114.
When working with vectors, it is important to recognize the difference between a vector and a
scalar (or real number). We know how to add, multiply, etc. when working with real numbers, but
have learned no such techniques for working with vectors. In fact, it is unclear as to exactly what
we mean when we say ”multiply a pair of vectors” or ”add a pair of vectors”. In this section, we
will see what it means to ”add” a pair of vectors together.
Definition 0.0.3. The sum of a pair of vectors ⃗u = ⟨u1 , u2 , u3 ⟩ and ⃗v = ⟨v1 , v2 , v3 ⟩ is given by the
sum of their components,
⃗u + ⃗v = ⟨u1 + v1 , u2 + v2 , u3 + v3 ⟩.
For example, with ⃗u = ⟨3, 1, 4⟩ and ⃗v = ⟨2, 5, 3⟩ we see that ⃗u + ⃗v = ⟨5, 6, 7⟩.
The definition of vector addition has a physical interpretation. Let’s add the vectors ⃗u = ⟨0, 7, 2⟩
and ⃗v = ⟨0, 1, 4⟩ to get ⃗u + ⃗v = ⟨0, 8, 6⟩.
We can construct the sum by placing the tail of ⃗v at the tip of ⃗u (or vice-versa):
5
Section 12.2
The vector that forms the third side of the triangle with ⃗v and ⃗u as the other two sides is
precisely ⟨0, 8, 6⟩.
This interpretation of adding a pair of vectors is known as the parallelogram law; the vector
that results from performing ⃗u + ⃗v is precisely the diagonal of the parallelogram whose sides are ⃗u
and ⃗v .
In addition to adding a pair of vectors, we have a type of multiplication operation on vectors:
we can multiply a vector by a scalar.
Definition 0.0.4. If c is a scalar (real number) and ⃗u = ⟨u1 , u2 , u3 ⟩, then the product c⃗u is given
by multiplying each component of ⃗u by c, that is
c⃗u = ⟨cu1 , cu2 , cu3 ⟩.
For any constant c, |c⃗u| = |c| · |⃗u|, where |c| is the absolute value of c.
Let’s look at an example: we’ll multiply ⃗u = ⟨0, 3, 4⟩ by 2 to get 2⃗u = ⟨0, 6, 8⟩:
6
Section 12.2
Notice that the direction of the vector is unchanged; we’ve simply stretched the vector by a
factor of 2.
If c < 0, the multiplication c⃗u will reverse the direction of ⃗u. For example, with ⃗u = ⟨0, 3, 4⟩,
−1 · ⃗u = ⟨0, −3, −4⟩ :
With these ideas in mind, it is not difficult to determine the physical interpretation of subtracting the vector ⃗v from ⃗u, i.e. calculating ⃗u − ⃗v ; multiplying ⃗v by −1 reverses its direction, and then
we add ⃗u and −⃗v , ⃗u + (−⃗v ):
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Section 12.2
Vectors are quite well-behaved with respect to addition and scalar multiplication; in fact, the
operations have the same properties as real number addition, as stated below:
Theorem 0.0.5. If ⃗a, ⃗b, and ⃗c are vectors and c and d are scalars, then:
• ⃗a + ⃗b = ⃗b + ⃗a
• ⃗a + (⃗b + ⃗c) = (⃗a + ⃗b) + ⃗c
• ⃗a + ⃗0 = ⃗a
• ⃗a + (−⃗a) = ⃗0
• c(⃗a + ⃗b) = c⃗a + c⃗b
• (c + d)⃗a = c⃗a + d⃗a
• (cd)⃗a = c(d⃗a)
• 1⃗a = ⃗a
Unit Vectors
A unit vector is a vector whose length is 1. The three unit vectors most often referenced are
the standard basis vectors ⃗i = ⟨1, 0, 0⟩, ⃗j = ⟨0, 1, 0⟩, and ⃗k = ⟨0, 0, 1⟩.
8
Section 12.2
Any vector ⃗u can be written as a sum of multiples of ⃗i, ⃗j, and ⃗k (called a linear combination of
⃗i, ⃗j, and ⃗k). For instance, the vector ⃗u = ⟨3, −2, 5⟩ can be rewritten as
⃗u = ⟨3, −2, 5⟩
= ⟨3, 0, 0⟩ + ⟨0, −2, 0⟩ + ⟨0, 0, 5⟩
= 3⟨1, 0, 0⟩ − 2⟨0, 1, 0⟩ + 5⟨0, 0, 1⟩
= 3⃗i − 2⃗j + 5⃗k.
In general, the vector ⃗u = ⟨u1 , u2 , u3 ⟩ can be written as
⃗u = u1⃗i + u2⃗j + u3⃗k.
To every vector ⃗u (except for ⃗0 = ⟨0, 0, 0⟩), we can associate a unit vector with the same direction
as ⃗u. We accomplish this by creating the vector
⃗u
|⃗u|
(in essence, dividing ⃗u by its length |⃗u|); since
1
|⃗
u|
is a scalar, the vector
⃗u
has length
|⃗u|
⃗u 1 = ⃗u = 1 |⃗u| = |⃗u| = 1.
|⃗u| |⃗u| |⃗u|
|⃗u|
⃗u
⃗u
is a unit vector. In particular,
points in the same direction as ⃗u, but is
|⃗u|
|⃗u|
scaled so that its length is 1.
In other words,
9
Section 12.2
Example
The vector
⃗u = ⟨−3, 0, 4⟩
has length
|⃗u| =
√
√
9 + 16 = 25 = 5,
thus is not a unit vector. However, the vector
3
4
⃗u
= ⟨− , 0, ⟩
5
5
5
is a unit vector since
√
√
⃗u 9
16
25
=
+
=
= 1.
5
25 25
25
The two vectors are graphed below; it is clear that dividing ⃗u by its length created a unit vector
pointing in the same direction as ⃗u:
10
Section 12.2
A Brief Digression on Notation:
Students often get the notation for absolute value of a real number, like | − 5|, confused with
the notation for the length of a vector, such as |⟨1, 3, 0⟩|. The interpretation of the meaning of the
”absolute value” bars depends entirely upon the context, i.e. the input. In the first case above, the
bars | · | enclose a number or scalar, so we interpret the bars as the absolute value of the scalar. In
the latter case, the bars enclose a vector. So the bars indicate that we need to find the length of
the vector.
There is really no difference between the two interpretations; finding the absolute value of a
scalar c amounts to determining the distance from the number 0 to c. This is really just the length
of the one-dimensional vector ⟨c⟩.
11