Let

The dot product  is equal to

.

The value of the dot product will return a number.  The formula for a dot product is:

Use the formula to find the dot product for the given vectors. 

To find the dot product, we first need to find the vectors in component form. This is done easiest with special right triangles, since their angles are 45 and 30 degrees.

First we can find the components of our first vector, :

The magnitude is 9, which means that we need to scale the triangle so that the hypotenuse is 9. Right now its value is 2, so we will have to multiply all of the side lengths by 4.5 to make that one be 9. This means that the components are and , so the component form of this first vector is

[note that although the picture has the 30-degree angle pointing towards the left, the vector should actuall be pointing to the right. A vector pointing left like that would have an angle listed as 150. Our 45-45-90 special right triangle will also have its 45-degree angle opening in the opposite direction from the vector]

Next we can find the component form of the vector by using the 45-45-90 special right triangle:

In this case the magnitude is 14, so we will have to scale this triangle so that the hypotenuse equals 14. , so we will multiply both legs by as well. This makes our component form be .

Finally, we will multiply the two vectors:

First rewrite the vectors in a bracket form 

 and 

.

The dot product

, 

Remember that when we take the dot product, we multiply the  components and the  components of the two vectors, and add them together. 

We can relate the dot product, length of two vectors, and angle between them by the following formula:

So the dot product of 

  and

 is the addition of each product of components:

now the length of the vectors of a and b can be found using the formula for vector magnitude:

So:

 hence  

First, we note that the dot product of two vectors is defined to be;

.

First, we find the left side of the dot product:

.

Then we compute the lengths of the vectors:

.

We can then solve the dot product formula for theta to get:

Substituting the values for the dot product and the lengths will give the correct answer. 

Solving the dot product formula for the angle between the two vectors results in the equation .

If we call the vectors a and b, finding the dot product and the lengths of the vectors, then substituting them into the formula will give the correct angle.

Substituting the values correctly will give the correct answer.

To find the angle between two vectors, use the following formula:

 is known as the dot product of two vectors. It is found via the following formula:

The denominator of the fraction involves multiplying the magnitude of each vector. To find the magnitude of a vector, use the following formula:

Now we have everything we need to find our answer. Use our given vectors:

So the angle between these two vectors is 102.09 degrees

We use the dot product to find the angle between two vectors. The dot product has two formulas:

We solve for the angle measure to find the computational formula:

Our vectors give dot products and lengths:

Substituting these values into the formula above will give the correct answer.