Professor Dave again, let’s find dot products.
We now understand what vectors are, and we can do some operations with them.
We can add and subtract them, and we can multiply them by scalars.
But we also want to be able to multiply two vectors together, and this is not as trivial,
because we have to understand what we mean by this operation.
There are actually two different ways that we can multiply two vectors, and the first
kind of multiplication will give us something called the dot product.
Let’s learn how to perform this operation.
The dot product is fairly intuitive.
If we have some vector, A, with components A one, A two, and A three, and another vector,
B, with components B one, B two, and B three, then its dot product will be denoted this
way, as A dot B.
This will be equal to A one B one, plus A two B two, plus A three B three.
It is very important to note that this is a scalar product.
The dot product does not give us another vector, it takes two vectors as an input, and gives
one number as an output.
For example, if A is two, four, and B is one, negative three, then A dot B is equal to (two
times one) plus (four times negative three), or two minus twelve, which gives us negative ten.
Because the dot product involves simple multiplication, it has very intuitive properties.
A dot B equals B dot A, because it doesn’t matter what order in which numbers are multiplied.
The distributive property also holds, and A dot the sum of B and C, or A dot the vector
that results from adding B and C, is the same as A dot B plus A dot C. A scalar times a
vector involved in a dot product is the same as the scalar multiplied by the dot product,
or the other combination of scalar and vector involved in the same dot product.
A dot A is simply the length of A, quantity squared.
And the zero vector dot A is the number zero, because we are just multiplying each component
of A by each component of the zero vector, which are all zeros, and then adding all of
those zeros up to get zero.
The dot product also crops up in an interesting theorem, which says that for two vectors,
A and B, A dot B is equal to the length of A times the length of B times the cosine of
the angle between them.
This means that if we want to calculate the angle between two vectors, we can simply bring
these vector lengths to the other side of the equation, and we get cosine theta as being
equal to A dot B divided by the product of the vector lengths.
This is one of the reasons the dot product is useful, as it gives us a great way to calculate
angles without doing any trigonometry.
Let’s give it a try.
What is the angle between vector A, with components two, two, negative one, and vector B, with
components five, negative three, two.
Well first, let’s find the length of each vector.
For A, that’s two squared, plus two squared, plus negative one squared, all under a square root.
We simplify to get nine, and then take the square root to get three.
For B, we get five squared, plus negative three squared, plus two squared, under the
square root.
We simplify to get thirty eight, and we can just leave this as root thirty eight.
Now let’s perform the dot product.
Multiplying the corresponding components, we get two times five, plus two times negative
three, plus negative one times two.
That’s ten minus six minus two, or two.
So plugging everything in, we get two on the top, and then three times root thirty eight.
That is equal to the cosine of the angle between the vectors.
To solve for the angle, we just take the inverse cosine, and since this is a pretty ugly number,
we just plug it into a calculator to get some decimal and press the inverse cosine button.
That gives us around 1.46 radians, which is about eighty four degrees.
This operation leads us to another definition.
We want to understand that if the angle between two vectors is half pi radians, or ninety
degrees, these vectors are orthogonal.
This pretty much means the same thing as being perpendicular to one another, and an important
thing to note is that any two orthogonal vectors will always give a dot product of zero, because
the cosine of half pi is zero.
This means that if you are computing a dot product and you get zero, those two vectors
are orthogonal.
So that’s what we need to know about vector dot products, which represent one way that
vectors can be multiplied.
The other way, called the cross product, is more complicated, and one interpretation will
require that we understand how to use matrices.
Matrices are the plural of the word matrix, which apart from being a super cool science-fiction
movie, is a construct that is ubiquitous in the study of linear algebra, so let’s move
forward and begin to study this new subject.
But before doing so, let’s check comprehension.