We're given vectors a and b,
and asked to find the dot product of vector a and b.
To find the dot product of two vectors,
we find the sum of the products
of the corresponding components.
And then if the dot product is positive, negative,
or zero, it does tell us something about the angle theta
between the two vectors.
Which we'll discuss more about after our first example.
So to find the dot product of vector a...
And vector b,
we would have negative three, times negative five,
plus two times two,
plus five times three.
So OK we would have negative three times negative five,
plus two times two,
plus five times three.
So we have 15 plus four plus 15,
which equals 34.
So our dot product equals 34.
And as I mentioned earlier,
the value of the dot product does tell us something
about the angle theta between the two vectors.
If the dot product is equal to zero,
then the two vectors are orthogonal or perpendicular,
meaning the angle theta between the two vectors
is equal to pi over two radians.
If the dot product is greater than zero or positive,
which is our case, then the angle theta between
the vectors is less than pi over two radians.
And if the dot product is negative,
then the angle theta between the new vectors
is greater than pi over two radians.
Again in our first example,
the dot product was positive,
so the angle theta between these two vectors
would be less than pi over two radians.
Let's verify that graphically.
If we were to graph our two vectors in space,
they would look like this.
And notice how the angle between these two vectors
is less than pi over two radians.
Let's take a look at a second example.
So we have the same question, just different vectors.
So we'd have vector a...
Dotted with vector b,
which would be equal to one times negative five,
plus negative four times six,
plus two times three.
So once again we have one times negative five,
plus negative four times positive six,
plus two times three.
So we have negative five,
plus negative 24,
plus six,
which is equal to negative 23.
Notice in this case the dot product is negative,
which means the angle between the two vectors, angle theta,
would be greater than pi over two radians.
Again let's verify this graphically.
Here are the graph of the two vectors,
and as we can see, the angle between the two vectors
is greater than pi over two radians.
We have an obtuse angle between these two vectors.
Let's take a look at one more example.
Again, same question, two different vectors.
So we have vector a...
Dotted with vector b.
So we'd have positive three times negative three,
plus negative four times positive four,
plus five times five.
So again, positive three times negative three,
plus negative negative four times positive four,
plus five times five.
So here we have negative nine,
plus negative 16,
plus 25,
which equals zero.
And again when the dot product equals zero,
the two vectors will be orthogonal or perpendicular,
and angle theta would be equal to pi over two radians.
Let's go ahead and verify this graphically as well.
Again here is the graph of the two vectors in space.
If we pause this at the right time...
We can see these two vectors are orthogonal
or perpendicular, which will always be the case
when the dot product equals zero.
I hope you found these three examples helpful.