In this post I’m going to look at something usually referred to as linear combinations. In the prior post, we had an occasion to look at coordinate systems where the axes are not at right angles. We saw that you can draw grid lines parallel to the axes; from any point, you can follow the grid lines to get the coordinates, in the same way you’d do if the axes are at right angles.
The graph above, from part 21, shows axes of guineas and shillings. The axes are not at right angles, but doing so wasn’t totally arbitrary: if you move one unit horizontally to the right, you add one pound to the value (a guinea is one pound plus one shilling). To find the value of Jane’s jar in terms of guineas and shillings, you would follow the grid lines, and see that it amounts to 4 guineas and 3 shillings.
We could interpret this same thing in terms of movement. If we start at the place where the axes meet (usually referred to as the origin), and we make 4 ‘guinea’ moves, and 3 ‘shilling’ moves, we arrive at the point representing Jane’s jar. Each of the 4 guinea moves is a move in a particular direction (here, diagonally up to the right) and of a particular distance. Each of the 3 shilling moves is a move in a particular direction (here, straight up) and of a particular distance.
From our starting point at the origin (meeting point of the axes), any combination of 4 guinea moves and 3 shilling moves will get us to the same ending point. We can take 1 guinea move, 1 shilling move, 3 guinea moves and finally 2 shilling moves, and end up at Jane’s jar. Or we could take 2 guinea moves, 1 shilling move, 1 guinea move, 2 shilling moves, and finally 1 guinea moves, and also end up at Jane’s jar. All that matters is how many guinea moves we did altogether, and how many shilling moves we did altogether. This can be expressed in shorthand, using a notation I haven’t used here before. Let’s write for a single guinea move, and for a single shilling move. Then we can write:
(a) Jane’s jar =
(b) Jane’s jar =
(c) Jane’s jar =
(d) Jane’s jar =
(e) Jane’s jar =
In the old tradition of pirate treasure maps, “go 100 paces North East,” each of the expressions in that list represents a particular path, though all happen to end up in the same place.
The above represents an informal introduction to the geometric notions of vectors, vector addition, and multiplication of vectors by a number. Textbooks often introduces vectors as arrows with direction and length, like our ‘guinea move’ or like our ‘shilling move’. They will tell you that to add two vectors, you need to draw the parallelogram with the two vectors as their sides, and then draw the diagonal of the parallelogram, and this diagonal represents the sum. They really are just talking about the result of two motions one after another.
In the diagram above, the blue arrow represents , the green one represents . In the box on the right, three different ways to combine 2 guinea moves and one shilling move are shown. Each path, if started at the origin, will end up in the same place. That place can be characterized as . When you focus on the geometry, and divorce yourself from the specific meaning of guinea values and shilling values, you can see that what we have shown here matches the textbook treatment of a vector as an arrow with direction and length (but with flexible starting point), and matches the textbook treatment of vector addition as the joining of two arrows end-to-start. Moreover, the idea generalizes to taking any number of the first vector, and any number of the second vector, and joining them all together, in what is known as a linear combination of the two original vectors.