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VECTORS LECTURE IN R2
A vector quantity is anything that is specied by both a magnitude and a direction.
by |v|=
x2 + y 2
x
y
the magnitude of v (denoted by |v|) is given
y
and the direction of v is determined through tan() = .
x
Given an algebraic vector v =
A geometric vector v with magnitude |v| and direction may be expressed as
v=
x
y
=
|v| cos()
|v| sin()
GEOMETRIC VECTORS IN R2
A scalar quantity is something that can be represented by a single number, for example
temperature or mass.
A vector quantity is anything that is specied by both a magnitude and a direction.
Examples of vector quantities are force and velocity.
We represent vector quantities simply as arrows with the magnitude of the vector
being the length of the arrow and the direction of the vector being the direction of the
arrow. On the printed page we represent vectors in bold and when written we use a
twiddle though it is often forgotten.
We will begin by considering vectors in two dimensional space (denoted R2 ) and then
move on to vectors in three dimensional space R3 .
Example 1: For the following vector v sketch 2v and 3v.
Geometric vectors are added using the strange parallelogram law of addition also called
tip to tail addition.
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Example 2: Three people are pulling on a pole. Person 1 pulls due north with a force
of f1 =20N. Person 2 pulls due east with a force of f2 =30N and Person 3 pulls NW with
a force of f3 =50N. Find the direction and magnitude of the combined force using tip to
tail addition.
|f1 + f2 + f3 | 55N
=
95
=
Obviously this geometric approach to vectors is very clumsy and although it might be
OK in physics, in mathematics we need much tighter control. We achieve this with the
simple but very fruitful mechanism of placing the vector in the x y plane so that it has
a nice home rather than living on the with streets no xed address. We may then simply
x
refer to the vector by its Cartesian coordinates v =
and the inaccurate approaches
y
above are replaced by simple vector algebra.
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Example 3: Sketch the vectors u =
2
5
and v =
4
1
in R2 .
Note that we could also write u = 2 , 5 and v = 4 , 1 . It doesnt really
matter whether we write the vectors vertically or horizontally.
Example 4:
a) Adopt an algebraic approach to nd u + v, u v and 2u + 3v where u and v
are as dened in Example 3.
b) Sketch u + v and u v in the diagram above, showing that the algebraic approach
to vector addition corresponds exactly to the geometric method of tip to tail addition.
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How do we convert algebraic vectors to geometric vectors in R2 ?
FACT
by |v|=
x2 + y 2
x
y
the magnitude of v (denoted by |v|) is given
y
and the direction of v is determined through tan() = .
x
Given a vector v =
Note that these are exactly the same denitions we used in complex numbers for |z |
and Arg(z )!!
2
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Example 5: Find the magnitude and direction of u =
and v =
.
5
1
u has magnitude
v has magnitude
4
29 and direction = 68 12
17 and direction = 165 58
We can also easily convert geometric vectors over to algebraic vectors.
A vector v in R2 with magnitude |v| and direction may be expressed as
FACT
v=
x
y
=
|v| cos()
|v| sin()
Note that this is just the usual conversion from polar to Cartesian form for complex
numbers!
Example 6: Convert the vectors in Example 2 to algebraic form and hence nd the
exact value of the magnitude of the combined force and its direction correct to the
nearest minute.
50
30
2
f1 + f2 + f3 =
50
20 +
2
50 2
50 2
(30 2 ) + (20 + 2 ) = 55.613N
|f1 + f2 + f3 | =
5.355
55.355
= 95 32
In the next lecture we will leave geometric vectors behind as we move into higher
dimensional spaces.
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