The vector u has length 19
and points at an angle of 30 degrees.
The vector v has length 11
and points at an angle of 40 degrees.
So I've already graphed the two vectors below,
where vector u is the blue vector,
where the length or magnitude is 19
and the direction is 30 degrees.
Vector v is the red vector, where the magnitude is 11
and the direction is 40 degrees.
Looking at the graph we can see
the angle between the two vectors
is equal to 40 degrees, minus 30 degrees or 10 degrees,
which is this small angle here.
And now there are two ways we can determine the dot product,
first vector u dotted with vector v
is equal to the magnitude of vector u
times the magnitude of vector v times cosine theta,
where theta is angled between the two vectors.
The second method would be to find the component form
of the two vectors shown above, then determine the sum
of the products of the x and y components
and we will go ahead and show both methods.
Let's first determine the dot product
using the angle between the two vectors,
which means vector u dotted with vector v
is equal to, again, the magnitude of vector u, which is 19
times the magnitude of vector v, which is 11
times cosine theta where theta is angled
between the two vectors, which is 10 degrees.
And now let's go to the calculator.
This is the Desmos scientific calculator.
The first step is to make sure the calculator
is in the degree mode, which it is.
For this calculator, we toggle between degrees
and radians by clicking here.
And now we enter 19 times 11
times cosine 10 degrees,
close parenthesis and enter,
to four decimal places, we have 205.8248.
And now to determine the dot product,
whereby the two vectors in component form shown here
and then determine the sum of the products
of the x and y components.
So for vector u the x component is equal
to the magnitude of vector u times cosine theta,
where theta here is the direction of the vector
and therefore the x component is equal
to 19 times cosine 30 degrees
and the y component is equal to the magnitude of vector u
times sine data, which gives us 19 times sine 30 degrees.
And now for vector v,
where the magnitude is 11 and the direction is 40 degrees,
the x component is 11 times cosine 40 degrees
and the y component is equal to 11 times sine 40 degrees.
So using the component form, vector u dotted with vector v
is equal to 19 cosine 30 degrees
times 11 cosine 40 degrees
plus 19 sine 30 degrees
times 11 sine 40 degrees.
And now let's just verify we get the same result
as we did above, going back to the calculator.
We have 19 cosine 30 degrees
times 11 cosine 40 degrees
plus 19 sine 30 degrees
times 11 sine 40 degrees.
And you can see, we get the same result, 205.8248.
I hope you found this helpful.