At the level of vectors there are also such «units». However, vectors not only have magnitude, but also direction. A unit vector specifies a «uniform direction». Not more.
In the following sections you will get to know how unit vectors are defined mathematically, what use they have and how you can construct unit vectors from given vectors.
The choice of the names «Hamilton» and «Gibbs» has a reason. You will find out about this in the course of the explanation.
Vectors, their coordinates and length – a quick review
Unit vectors are vectors with a specific property related to their length. That is, to understand the concept of unit vectors, you need an understanding of vectors in general and how to determine their length.
For the details of vectors, the «linear combination» explanation is recommended. There you will learn what a vector actually is (spoiler: vectors are much more than arrows).
vectors as arrows
For both points there is a repetition of the most important statements that are necessary for the explanation in this section.
vectors as arrows | Notation for vectors
A vector is an arrow in the plane or in space. Such an arrow is clearly characterized by its length and direction.
An «arrow over the name» is used to indicate that a given object is a vector. For example, if the object \(a\) is a vector, it is highlighted with \( \vec{a}\).
For completeness: Vectors can also live in other places than plane or space. For example, there are also vectors along the number line or vectors in a four-dimensional location. The following is restricted to the plane case. The same statements apply to arrows in space.
You can imagine an arrow with length and direction relatively easily (see figure 1). But you can’t really count on that. This applies in particular to the length of vectors. The remedy here is the introduction of a coordinate system.
Figure 1: Vectors as arrows can be visualized well.
Construction of a coordinate system – first contact with unit vectors
You construct a coordinate system as follows: You choose a point that you origin call From this origin you choose two directions for the plane (for space it would be three directions).
You have only one restriction for the choice of directions: They must not lie on the same straight line through the origin (see Figure 2).
Figure 2: The selected directions must not lie on this origin line.
Now all you have to do is decide what it means to move one «unit» along the given directions. You can define a separate «unit» for each direction.
Directions and units for the plane
For example, you can choose «South» and «Northeast» as two directions for the plane. Along the south direction, «one unit» means you’ve moved an inch; along the northeast direction, however, by one meter (see Figure 3).
Figure 3: Concrete specification of two directions and «units» for the plane.
If it says: You are moving three units south and four units north-east, then you have gone three centimeters south and then four meters north-east (or vice versa).
In the language that is gradually being developed here, you have two levels from the origin linearly independent unit vectors predetermined. These unit vectors define your coordinate system.
You can let your creativity run free, except for the restrictions mentioned. However, there are certain options that are particularly easy to manage. Historically, the first choice was a coordinate system now known as Cartesian coordinate system referred to as.
That Cartesian coordinate system for the level consists of two directionseach perpendicular to each other stand. And for each direction, «one unit» means the same thing.
You can then assign two numbers to each point on the plane, which tells you how many times you have to go «right» or «up» to reach the point from the origin.
Cartesian coordinate system of the plane and specification of points
In Figure 4 is the Origin O highlighted and two other points. Since you do not have to move from the origin to reach the origin, it is assigned the two numbers 0 and 0.
This is written more compactly as (0|0). This notation contains the agreement that the first number describes the steps in the «right» direction and the second number describes the steps in the «up» direction. These two numbers form the coordinates of the origin with respect to the selected coordinate system.
Figure 4: Plane with a Cartesian coordinate system.
For the point a you have to walk two steps «right» and three steps «up». It therefore has the coordinates (2|3).
The Point B you can reach it by moving three steps «left» and 5 steps «down». In order not to introduce new directions, negative numbers are used. So three steps to «left» is the same as «minus three steps» to «right». So the point B has the coordinates (-3|-5).
To make the Cartesian coordinate system useful for vectors, the following important agreement hit: You move each vector so that its beginning is at the origin without changing its direction (Figure 5). Such a shift is also called parallel displacement.
Figure 5: Parallel displacement of the blue vector to the pink vector at the origin. The vertex of the pink vector has coordinates (x|y).
After you do that, the tip of the vector will end up at a point with coordinates \( (x \mid y) \).
Identify vectors with dots | Notation for vectors by coordinates
You can do this with any vector. Instead of talking about arrows geometrically, a vector with the coordinates of its peak identified within a Cartesian coordinate system.
If the vertex of the vector \(\vec{a}\) has the coordinates \((x \mid y)\), then this identification is carried out by
\
written down. The payment x and y they are also called components of the vector \(vec{a}\) with respect to the chosen coordinate system.
Don’t forget that there is a vector on the left: a vector is a geometric object; it doesn’t care what coordinate system you use. On the right, however, there are two numbers that are strongly dependent on your choice of a coordinate system. For example, if you turn your head to the right and with it the two directions of your coordinate system, the coordinate system will change and with it the two numbers on the right. The vector, however, remains unchanged. Everything that follows from now on will depend on the choice of the coordinate system.
You have assigned two numbers to each vector. And you can count with numbers.
Identify vectors with dots | add vectors
Consider Figure 6. The top of the vector \(vec{a}\) is at \((2 \mid 5)\) and you have
\
Figure 6: Vectors are identified with the points of their vertices.
In the case of the vector \(\vec{b}\), on the other hand, the peak is at \((4 \mid 3\), i.e
\
Now you can calculate with these two vectors. For example, you can add the two vectors together
\
Vectors can be added and multiplied by a number. In the Cartesian coordinate system you can also determine the length of a vector with one of the oldest theorems of mathematics.
length of vectors
See Figure 7 for this. You can understand the vector as the hypotenuse of a triangle whose two legs are exactly the horizontal and vertical distance of the vector tip from the origin.
Figure 7: To determine the length of a vector, it becomes the hypotenuse of a triangle.
triangles, hypotenuses, catheti; that sounds a lot like that Pythagorean theorem. And that is exactly the case.
Length of a vector in a Cartesian coordinate system
Let \(\vec{a}\) be a vector with
\
the length of \(\vec{a}\), notated as \(|\vec{a}|\), in a Cartesian coordinate system is given by
\
You can also find the designation for the length of a vector standard. The norm on the Pythagorean theorem is then called more concrete Euclidean norm. Instead of double dashes «» often only single dashes «» are used. However, this can lead to confusion with the absolute value of a number. If you had a vector of space, you would only have to add \(z^2\) to the formula; So you would have \(|\vec{a}|=\sqrt{x^2+y^2+z^2}\).
So in order to determine the length of a vector in a Cartesian coordinate system, you only need to insert its coordinates into the definition formula.
Determine lengths of concrete vectors
You have the two vectors again
\( \vec{a}=\left(\begin{array}{c}2 \\ 5 \end{array} \right) \) and \(\vec{b}=\left( \begin{array} {c} 4 \\ 3 \end{array}\right)\)
The corresponding lengths are then
\
and\
for their sum vector
\
you have however
\
First insight into the triangle inequality
Add up the two individual lengths of the vectors
\
A comparison with the length of the sum vector then shows you that
\is applicable.
The property
\| + |\vec|\]
will as triangle inequality (see Figure 8). This is a fundamental property required of all norms within mathematics.
Figure 8: Triangle inequality illustrated in the plane.
This completes the repetition. As a short summary:
-
With the agreement that vectors should always start at the origin, the vectors were identified with the coordinates of their vertices within a Cartesian coordinate system.
The enormous importance of the Pythagorean theorem
If you provide the plane or space with a Cartesian coordinate system, you will inevitably be greeted by the Pythagorean theorem.
And that is exactly what is done in many disciplines: from architecture to physics and construction to game development; everywhere the plane or space is provided with a Cartesian coordinate system. And of necessity the Pythagorean theorem presents itself whenever lengths or angles are desired.
The Term «Vector» and Vector Notations
Sir William Rowan Hamilton introduced the term «vector» in about 1843. Josiah Willard Gibbs introduced notations about 40 years later that are still used today in all mathematics in the context of vectors. These include in particular the notation for the dot product and cross product two vectors.
Unit vector definition and examples
First of all, a unit vector is a vector. The only peculiarity that makes it a unit vector is its length.
unit vector – a vector of length 1 | Notation for unit vectors
Let \(\vec{a}\) be a vector. Then \(\vec{a}\) unit vector, if its length is 1; so if applies
\
To emphasize that \(\vec{a}\) is a unit vector, the notation \
used, read as «a hut».
So in order to be able to talk about unit vectors at all, you have to define a length for vectors in advance.
In the language of…