7.2   MORE ON VECTORS

What does a vector actually “mean”? The answer has to be “anything you
like.” Anything, that is, that cannot be represented by a single number but
requires a string of numbers to define it. It could even be a shopping list:

can be written in matrix format as

which we might write in a line of text as (orange, lemon, grapefruit) (5,3,2)′
or else place the dot between them that we use for scalar product. The
numbers on the right have defined a “mixture” of the items on the left.

Rather than fruit, we are more likely to apply vectors to coordinate
systems—but we are still just picking from a list.

We might define i, j, and k to be unit vectors all at right angles, say, east,
north, and up. We can call them basis vectors.

When we say that point P has coordinates (2,3,4)′, we mean that to get
there, you start at the origin and go 2 m east, then 3 m north, and 4 m up.

We could write this as

which is a mixture of the basis vectors defined by a matrix multiplication—
vectors are just skinny matrices.

Now, when we turn our minds to applications, we can see many uses for
vector operations. When a force F moves a load a distance x, the work done
is given by their scalar product F · x.

As before, we take products of corresponding elements and add them up,
to get a scalar number.

We usually think in terms of “the matrix multiplies the vector.” But how
about thinking of the vector multiplying the matrix? What does it do to it?
Consider the following matrix:

From one perspective, the top element is equal to the scalar product of the
top row of the matrix with the vector (x,y,z)′. Similarly, the other elements
are the scalar products of the vector with the middle and bottom rows of the
matrix, respectively.

So we have

But there is another way of seeing it. The answer is the same as

So we also have

Suppose that point P is defined in terms of a second set of basis vectors, u, v,
and w, so that its coordinates (x,y,z)′ mean xu + yv + zw. To fi nd the coordinates
in terms of i, j, and k, we simply multiply and add up the contributions
from u, v, and w.

We can “transform the coordinates” by multiplying (x,y,z)′ by a matrix
made up of columns representing vectors u, v, and w, to end up with a vector
for P as a mixture of i, j, and k.