We want to find the dot product
of vector a to b graphed below.
So let's call the blue vector, vector a,
and the red vector, vector b.
Because both vectors have initial points at the origin,
zero comma zero,
and vector a has a terminal point
at the point
three comma negative four,
and vector b has its terminal point at the point
negative two comma five.
Vector a in component form
would have an x component of positive three,
and a y component of negative four.
And vector b
would have an x component of negative two
and a y component of positive five.
To find the dot product of vector a and b,
we find the sum of the products
of the corresponding components.
And vector v dotted with vector w
is equal to vector w dotted with vector v.
And therefore, the order of the dot product
does not affect the result.
So if I call this vector a and this vector b,
we would still get the same dot product.
So vector a
dotted with vector b
will be equal to
three times negative two
plus negative four times five.
So again, three times negative two,
plus negative four times positive five
is equal to negative six plus negative 20,
which is equal to negative 26.
And the value of the dot product
does tell us something about the angle theta
between the two vectors.
Looking at these two vectors on the coordinate plane,
the angle theta between these two vectors
would be this angle here.
So one of the properties of the dot product is
that if the dot product is equal to zero,
the two victors are orthogonal or perpendicular,
and theta's equal to pi over two radians.
If the dot product is positive or greater than zero,
then the angle theta is less than pi over two radians.
And the third case is when the dot product
is negative or less than zero,
then the angle theta is greater than pi over two radians.
So going back to our example,
notice our dot product is negative,
or less than zero.
Which indicates theta is greater than pi over radians.
Which we can see from the graph is the case.
We have an obtuse angle between vector a and vector b.
Let's take a look at one more example.
We have the same question, just two different vectors.
So we'll call a blue vector vector a again,
and we'll call the red vector vector b.
Again, both initial points of the vectors
are at the origin,
zero comma zero.
The terminal point of the blue vector
is the point negative four comma two.
The red vector has its terminal point
at the point two comma four.
Which means vector a in component form
would have an x component of negative four,
a y component of positive two.
And vector b
would have an x component of positive two
and a y component of positive four.
So let's find vector a dotted with vector b.
So vector a
dotted with vector b.
Again, the order of the dot product doesn't matter.
The result would be the same.
So our dot product here
will be equal to negative four times positive two
plus two times four.
So once again, we would have negative four
times positive two
plus two times four.
Which is equal to negative eight plus eight,
which equals zero.
Remember when the dot product equals zero,
the two vectors are orthogonal or perpendicular,
and theta is equal to pi over two radians.
And looking at the graph,
this does verify our result.
The angle theta here is pi over two radians.
And therefore, these two vectors are orthogonal,
or perpendicular.
I hope you found these two examples helpful.