76 2. PARAMETRIC, POLAR, AND VECTOR FUNCTIONS
2.3 VECTOR FUNCTIONS
One of the most unsatisfying things about the treatment of position, velocity, and acceleration
in single-variable calculus was that these things happened in one dimension. is situation can
be amended by using vectors. A vector is a lot like a point – it has coordinates and a dimension
– but a vector is thought of as specifying a direction and a magnitude rather than just a point in
space.
Example 2.54 A vector:
(0,4)
(0,-4)
(4,0)(-4,0)
e vector Ev D .2; 3/
(2,3)
We use a small arrow over a variable name to denote that an object is a vector. So, the vector
shown above is Ev D .2; 3/. In a way, vectors are like multi-dimensional numbers that let us
work in as many dimensions as we need to. Vectors have their own arithmetic. It’s called vector
arithmetic.
Knowledge Box 2.7
Vector Arithmetic
If Ev D .v
1
; v
2
; : : : ; v
n
/ and Ew D .w
1
; w
2
; : : : ; w
n
/ are vectors in n-dimensions
and c is a constant, then we define the following.
• c Ev D .cv
1
; cv
2
; : : : ; cv
n
/ (scalar multiplication)
• Ev C Ew D .v
1
C w
1
; v
2
C w
2
; : : : ; v
n
C w
n
/ (vector addition)
• Ev Ew D .v
1
w
1
; v
2
w
2
; : : : ; v
n
w
n
/ (vector subtraction)
• Ev Ew D v
1
w
1
C v
2
w
2
C : : : Cv
n
w
n
(dot product)
2.3. VECTOR FUNCTIONS 77
Example 2.55 If Ev D .1; 2; 1/ and Ew D .0; 2; 4/, compute 5 Ew Ev.
Solution:
5 .0; 2; 4/ .1; 2; 1/ D .0; 10; 20/ .1; 2; 1/ D .0 1; 10 2; 20 .1// D .1; 8; 21/
˙
Example 2.56 If Ev D .2; 1; 3/ and Ew D .1; 2; 1/, compute Ev Ew.
Solution:
Ev Ew D 2 1 C 1 2 C 3 1 D 2 C 2 C3 D 3
˙
ere are a number of useful algebraic rules for vector arithmetic.
Knowledge Box 2.8
Vector Algebra
• c .Ev C Ew/ D c Ev C c Ew
• c .d Ev/ D .cd / Ev
• Ev C Ew D Ew C Ev
• Eu .Ev C Ew/ D Eu Ev C Eu Ew
Definition 2.4 e length or magnitude of a vector Ev D .v
1
; v
2
; : : : ; v
n
/, denoted jEvj, is given by
jEvj D
q
v
2
1
C v
2
2
C C v
2
n
Notice that this is similar to the formula for calculating distance. e length of a vector is the
distance it spans, so this is a natural definition.
It turns out that there is a really useful property of one way we multiply two vectors, the dot
product in Knowledge Box 2.7, that also uses the length of vectors. Suppose we have two vectors
Ev and Ew as shown in Figure 2.8. en, we can use the dot product together with the lengths of
the vectors to calculate the angle between them.
78 2. PARAMETRIC, POLAR, AND VECTOR FUNCTIONS
Ev
Ew
Figure 2.8: Two vectors.
Knowledge Box 2.9
Formula for the angle between vectors
cos./ D
Ev Ew
jvjjwj
What makes this property so useful? If two vectors are at right angles, then the cosine of the
angle between them is zero. is means we can detect right angles in a new and easy way and in
any number of dimensions. In particular, two vectors are at right angles to one another if and
only if their dot product is zero.
Example 2.57 Which of the following pairs of vectors are at right angles to one another?
1. Ev D .1; 1/ and Ew D .1; 1/
2. Es D .1; 2; 1/ and
E
t D .1; 2; 3/
3. Ea D .3; 1; 2/ and
E
b D .2; 2; 3/
2.3. VECTOR FUNCTIONS 79
Solution:
Compute dot products:
E
v
E
w
D
.1; 1/
.1;
1/
D
1
1
D
0
So Ev and Ew are at right angles to one another.
Es
E
t D .1; 2; 1/ .1; 2; 3/ D 1 C4 3 D 0
So Es and Er are at right angles to one another.
Ea
E
b D .3; 1; 2/ .2; 2; 3/ D 6 2 C 6 D 10
So Ea and
E
b are not at right angles to one another.
˙
Definition 2.5 Objects at right angles to one another are also said to be orthogonal to one another.
So with our new word we can re-state this property of the dot product as follows. Two vectors
are orthogonal if and only if they have a dot product of zero.
Definition 2.6 e cross product of Ev D .v
1
; v
2
; v
3
/ and Ew D .w
1
; w
2
; w
3
/ is defined to be:
Ev Ew D . v
2
w
3
v
3
v
2
; v
3
w
1
v
1
w
3
; v
1
w
2
v
2
w
1
/
e cross product of two vectors is only defined in three dimensions. It is a very special purpose
operator, but it is quite useful in physics. e following property is used to construct systems of
directions.
Knowledge Box 2.10
Orthogonality of the cross product
If Ev and Ew are non-zero vectors in three dimensions that are not scalar
multiples of one another, then Ev Ew is at right angles to (orthogonal to)
both Ev and Ew.
80 2. PARAMETRIC, POLAR, AND VECTOR FUNCTIONS
Example 2.58 Find a vector at right angles to both (1, 1, 2) and (3, 0, 4). Check your answer
by taking the relevant dot products.
Solution:
Use the cross product:
.1; 1; 2/ .3; 0; 4/ D .1 4 2 0; 2 3 1 4; 1 0 1 3/ D .4; 2; 3/
Check:
.1; 1; 2/ .4; 2; 3/ D 4 C 2 6 D 0
p
.3; 0; 4/ .4; 2; 3/ D 12 C 0 12 D 0
p
So the vector .4; 2; 3/ is orthogonal to .1; 1; 2/ and .3; 0; 4/.
˙
Definition 2.7 A unit vector is a vector Ev with jEv j D 1.
is is an almost trivial definition, but, coupled with the next Knowledge Box rule, it captures
an important feature of vectors.
Knowledge Box 2.11
e unit vector in the direction of a given vector
If Ev is not zero, then
1
jEvj
Ev
is the unit vector in the direction of Ev.
Unit vectors have a number of applications, but the one we will use the most often is that they
capture a notion of direction. ere are an infinite number of vectors in any given direction, but
only two unit vectors, and they point in opposite directions from one another.
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