Coordinate system

The geometrical interpretation of vectors depends on the coordinate system in which the vectors are represented. Throughout this section we have used the x, y, and z axes, and we’ve described vectors as components along each of these directions. This is a very convenient coordinate system; we have a set of three perpendicular axes, and a set of three unit vectors {ˆı, ˆ, ˆk} that point along each of the three axis directions. Every vector is implicitly defined in terms of this coordinate system. When you and I talk about the vector ~v = 3ˆı+ 4ˆ+ 2ˆk, we are really saying, “start from the origin (0, 0, 0), move 3 units in the x-direction, then move 4 units in the y-direction, and finally move 2 units in the z-direction.” It is simpler to express these directions as ~v = (3, 4, 2), while remembering that the numbers in the bracket measure distances relative to the xyz-coordinate system.

It turns out, using the xyz-coordinate system and the vectors {ˆı, ˆ, ˆk} is just one of many possible ways we can represent vectors. We can represent a vector

~v as coefficients (v1, v₂, v₃) with respect to any basis {ˆe1, ˆe₂, ˆe₃} as follows: ~v = v1eˆ1+ v2ˆe2+ v3eˆ3. What is a basis, you ask? I’m glad you asked, because this is the subject of the next section.

3.3 Basis

One of the most important concepts in the study of vectors is the concept of a basis. Consider the space of three-dimensional vectors R³. A basis for R³ is a set

Basis of vectors {ˆe1, ˆe2, ˆe3} which can be used as a coordinate system for R³. If the set of vectors {ˆe1, ˆe₂, ˆe₃} is a basis, then you can represent any vector ~v ∈ R³ as coefficients (v1, v₂, v₃) with respect to that basis:

~

v = v1ˆe1+ v2eˆ2+ v3ˆe3.

The vector ~v is obtained by measuring out a distance v1 in the ˆe1 direction, a distance v2 in the ˆe2 direction, and a distance v3 in the ˆe3 direction.

You are already familiar with the standard basis {ˆı, ˆ, ˆk}, which is associated with the xyz-coordinate system. You know that any vector ~v ∈ R³can be expressed as a triplet (vx, v_y, v_z) with respect to the basis {ˆı, ˆ, ˆk} through the formula

~v = vxˆı + vyˆ + vzˆk. In this section, we’ll discuss how to represent vectors with respect to other bases.

An analogy

Let’s start with a simple example of a basis. If you look at the HTML code behind any web page, you’re sure to find at least one mention of the colour stylesheet directive such as background-color:#336699;. The numbers should be inter-preted as a triplet of values (33, 66, 99), each value describing the amount of red, green, and blue needed to create a given colour. Let us call the colour described by the triplet (33, 66, 99) CoolBlue. This convention for colour representation is called the RGB colour model and we can think of it as the RGB basis. A basis is a set of elements that can be combined together to express something more complicated.

In our case, the R, G, and B elements are pure colours that can create any colour when mixed appropriately. Schematically, we can write this mixing idea as

CoolBlue = (33, 66, 99)_RGB = 33R + 66G + 99B,

where the coefficients determine the strength of each colour component. To create the colour, we combine its components as symbolized by the + operation.

The cyan, magenta, and yellow (CMY) colour model is another basis for repre-senting colours. To express the “cool blue” colour in the CMY basis, you will need

Vector products the following coefficients:

(33, 66, 99)RGB = CoolBlue = (222, 189, 156)CM Y = 222C + 189M + 156Y.

The same colour CoolBlue is represented by a different set of coefficients when the CMY colour basis is used.

Note that a triplet of coefficients by itself does not mean anything unless we know the basis being used. For example, if we were to interpret the triplet of coordinates (33, 66, 99) with respect to the CMY basis, will would obtain a completely different colour, which would not be cool at all.

A basis is required to convert mathematical objects like the triplet (a, b, c) into real-world ideas like colours. As exemplified above, to avoid any ambiguity we can use a subscript after the bracket to indicate the basis associated with each triplet of coefficients.

Discussion

It’s hard to over-emphasize the importance of the basis—the coordinate system you will use to describe vectors. The choice of coordinate system is the bridge between real-world vector quantities and their mathematical representation in terms of components. Every time you solve a problem with vectors, the first thing you should do is draw a coordinate system, and think of vector components as measuring out a distance along this coordinate system.

3.4 Vector products

If addition of two vectors ~v and ~w corresponds to the addition of their components (v_x+w_x, v_y+w_y, v_z+w_z), you might logically think that the product of two vectors will correspond to the product of their components (vxw_x, v_yw_y, v_zw_z), however, this way of multiplying vectors is not used in practice. Instead, we use the dot product and the cross product.

The dot product tells you how similar two vectors are to each other:

~

v · ~w ≡ v_xw_x+ v_yw_y + v_zw_z ≡ k~vkk ~wk cos(ϕ) ∈ R,

Vector products where ϕ is the angle between the two vectors. The factor cos(ϕ) is largest when the two vectors point in the same direction because the angle between them will be ϕ = 0 and cos(0) = 1.

The exact formula for the cross product is more complicated so I will not show it to you just yet. What is important to know is that the cross product of two vectors is another vector:

~v × ~w = {a vector perpendicular to both ~v and ~w } ∈ R³.

If you take the cross product of one vector pointing in the x-direction with another vector pointing in the y-direction, the result will be a vector in the z-direction.