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We begin with multiplication
of a vector by a scalar.

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When you multiply a
vector, A, by a scalar,

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this multiplicative factor
just rescales the magnitude

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or the length of the vector.

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Let us look at
the vector 2 times

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A. This is in the same
direction as the vector A,

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but is twice as long.

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This is vector B.
A vector is defined

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by its magnitude and direction.

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So this vector B is the
same anywhere in space,

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including at the origin.

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If I want minus 0.5
times B, this vector

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is in the opposite direction
of B and is half the length.

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Now let's look at
vector addition.

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Here's a vector A. Here is B.
How do we add them graphically?

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We slide the tail of
B to the head of A.

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And their sum is a vector
drawn from the tail of A

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to the head of B. I could
have also added A to B

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by sliding the tail
of A to the head of B.

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You can see that this
makes a parallelogram,

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and the sum, vector C,
is just the diagonal

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of this parallelogram.

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Subtraction can be thought
of as just multiplication

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and addition.

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If I have C is
equal to A minus B,

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I just need to add A to
the vector minus B. Minus B

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is negative 1 times B,
which is this vector here.

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Now I only have to
add A to minus B.

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Let's do another example.

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Here are my vectors A and B
do not start at the origin.

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But since vectors are the
same anywhere in space,

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I can go through
the process here.

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I want A minus B. So I
first multiply B by minus 1

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to find minus B. And then
I move the tail of minus B

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to the head of A and
add the two like this.