PROFESSOR: I now want to discuss the
subtraction of vectors.
If we have the same vectors A and B and C now equals A minus
B, then it should not come as a surprise that A minus B
equals Ax minus Bx in the x-direction plus Ay minus B of
y in the y-direction plus A of z minus B of z in the
z-direction.
So this is the x-component of the vector C, this is the
y-component, and this is the z-component of the vector C.
We can also see that in a geometrical way, to get some
feeling for it, what it means to subtract vectors.
And again, I will take a two-dimensional case.
It's easier to see.
And for simplicity, I will take the vectors only in the
y- and the x-plane.
But what I do holds in general, also in three
dimensions.
So we have now here the y-axis, and we have the plus
x-axis, the origin plus y-axis.
Let this vector be A and let this vector be B. And we want
to know what the vector C is, which is A minus B.
Now I can construct a parallelogram by drawing this
line from B to A. Draw a parallel line through the tail
of B. Draw a parallel line through the tip of A parallel
to B. And where these two intersect, this is now my
vector C, which is A minus B.
You can easily see this in a different way.
You can see that this is, indeed, correct.
I can think of this subtraction as the following:
C plus B equals A. So the question now is, which vector
do I have to add to B, namely, the vector C, to get A?
Well, you see immediately that if I add this vector to this
vector, following the rules of the adding of vectors making a
parallelogram, that this vector A is the sum of B and
C. That's one way of doing it.
Another way of doing it is saying, ah-ha!
C is also A plus minus B. So minus B is this vector, it's
the same as this one, but 180 degrees flipped over.
Now I have to add this vector to A to find C.
Well, there we go.
I make a parallelogram.
There we go.
And where do I end up?
Right here.
So notice that the sum of this vector and this vector
according to the summation rules indeed
gives me this vector.
And so this is a nice way of sort of seeing in your head
geometrically what it means when you subtract vectors.