The Unit Components

Usually, when working in 2 or 3 dimensions we use the standard unit component vectors i,j,k. These are vectors of length 1 with i in the direction of the x axis, j in the direction of the y axis and k in the direction of the z axis as shown below.

Writing vectors in component form

If we have a given vector v imagine the start of v being at the origin. Then suppose the other end has co-ordinates (a,b,c). That is if we travel distance a along the x axis, b along the y axis and c along the z axis we will have travelled along v. However, if you remember vector addition from the introduction to vectors you will notice that moving along a certain distance in one direction and then a certain distance in another is really just adding vectors.

So moving a along the x axis is moving a in the direction of i and since i is of length 1 we have really done ai. If we now move b along the y axis (the direction of j) we have relay just added bj to ai. Again if we move along the z axis by c we add ck. So we can now write

v = ai + bj + ck

mple of the vector 3j+2i in component form

Column and Row Vectors

When we have a vector in component form we can then write it as a column or a row vector. To do this we just write

or

Why have we done this? (addition)

Writing in component form may seen an odd thing to do but it is very useful. We now have some real numbers to work with (a,b,c) so we will find working with vectors simpler. For example addition is made easier as we can just add the components.

Example of addition

If v = 2i + 3j + k and u = -i + 3j then

v + u = (2-1)i + (3+3)j + (1+0)k= i +  6j + k

Components are also useful because they have some meaning. The tell us what the vector looks like in our normal co-ordinate system making it easier to understand what a vector “looks like”.

Finding the Norm or Size of a vector in component form

One of the benefits of the component form is it allows us to work out the norm or size of the vector. This is done using Pythagoras theorem since the components form a right angled triangle.

If v = ai + bj + ck then the norm of v is

Also see

• An introduction to vectors (0)
• Vectors: Dot or Scalar Product (0)
• Log Laws: Adding and Subtracting Logs (0)
• Basic Arithmatic in Binary (1)

Note// Whilst when first being introduced to vectors we usually are dealing with some sort of physical space, such as a plane, vectors can actually be used much more generally for example functions actually form and infinite dimensional vector space and colours are really just an example of another 3D vector space.

To start with vectors can be thought of as “arrows” on either a plane or in space (from the more general definition of vectors you might use later you’ll find that you can use any set of “axis” or bases  such as functions). The magnitude of a vector is represented by the length of the line whilst the direction is by the direction of the arrow.

Lets consider the 2D plane.

Drawing Vectors

The vector a is drawn an arrow in order that we can show the direction. The length of the arrow indicated the magnitude. By reversing the the direction of the arrow we can get -a.

Equality of Vectors

Two vectors are equal if they have the same direction and size or magnitude. This means that when we represent vectors using arrows two vectors can be equal even if they are drawn in different places provided they are parallel, pointing the same way and of the same length.

Addition of Vectors in the Plane

We can also add vectors together. To do this we “join” them so that we can see the combination of where they are heading. This means that adding vectors changes both the direction and the magnitude of the vector.

Multiplication of a Vector by a Scalar

And finally we can multiply a vector by a scaler.

This only changes the magnitude of the vector. In the case in the diagram multiplying a vector by 2 doubles its magnitude but keeps the direction the same.

More complex things can be done with vectors, such as the dot product, and they can also be written down numerically so that computations can be done with them which will be shown in later lessons.

Also see

• Writing Vectors in component form (0)
• Vectors: Dot or Scalar Product (0)
• Log Laws: Adding and Subtracting Logs (0)
• Matrix Calculator, C++ Program by David Woodford (5)
• Basic Arithmatic in Binary (1)

The dot product is defined to be

where is the angle between the vectors a and b.

Things to notice:

1. if a and b are parallel so the dot product is simply the product of their lengths
2. if and and b are perpendicular so the dot product is 0
3. 
4. if a is a unit vector, is |a|= 1 then the dot product gives the magnitude of the component of b in the direction of a

Dot product in component form (with i,j,k’s)

The dot product is very easy to use in component form because of 1 and 2. Since i,j and k are all perpendicular to each other:

and since any vector is parallel to itself and i,j and k are all unit vectors

hence we can find the dot product of with
Now we simply take the dot product of each term in a with each term of b, in a similar way to how you multiply out brackets, to get

and substituting in the above equations for the dot products of i,j,k we get

Also see

• Writing Vectors in component form (0)
• An introduction to vectors (0)