Since vectors have sign and magnitude they can’t be written down as simply a a scalar (number). They can of course be given a symbol that we know represents a given vector, as we did in the introduction to vectors, but these can be difficult to work with. A common way is to use component form. In this form a vector is given as a sum of vectors along the axes and so we can easy work with them.

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