We begin with multiplication of a vector by a scalar.
When you multiply a vector, A, by a scalar,
this multiplicative factor just rescales the magnitude
or the length of the vector.
Let us look at the vector 2 times
A. This is in the same direction as the vector A,
but is twice as long.
This is vector B. A vector is defined
by its magnitude and direction.
So this vector B is the same anywhere in space,
including at the origin.
If I want minus 0.5 times B, this vector
is in the opposite direction of B and is half the length.
Now let's look at vector addition.
Here's a vector A. Here is B. How do we add them graphically?
We slide the tail of B to the head of A.
And their sum is a vector drawn from the tail of A
to the head of B. I could have also added A to B
by sliding the tail of A to the head of B.
You can see that this makes a parallelogram,
and the sum, vector C, is just the diagonal
of this parallelogram.
Subtraction can be thought of as just multiplication
and addition.
If I have C is equal to A minus B,
I just need to add A to the vector minus B. Minus B
is negative 1 times B, which is this vector here.
Now I only have to add A to minus B.
Let's do another example.
Here are my vectors A and B do not start at the origin.
But since vectors are the same anywhere in space,
I can go through the process here.
I want A minus B. So I first multiply B by minus 1
to find minus B. And then I move the tail of minus B
to the head of A and add the two like this.