# Precalculus : Matrices and Vectors

## Example Questions

### Example Question #1 : Find The Dot Product Of Two Vectors

Find the dot product of the two vectors

and

.

Explanation:

To find the dot product of two vectors,

find the product of the x-components,

find the product of the y-components,

and then find the sum.

### Example Question #2 : Find The Dot Product Of Two Vectors

Evaluate the dot product of the following two vectors:

Explanation:

To find the dot product of two vectors, we multiply the corresponding terms of each vector and then add the results together, as expressed by the following formula:

### Example Question #3 : Find The Dot Product Of Two Vectors

Let

Find the dot product of the two vectors

.

Explanation:

Let

The dot product  is equal to

.

### Example Question #4 : Find The Dot Product Of Two Vectors

Let

Find the dot product of the two vectors

.

Explanation:

Let

The dot product  is equal to

.

### Example Question #5 : Find The Dot Product Of Two Vectors

Determine the dot product of  and .

Explanation:

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.

### Example Question #6 : Find The Dot Product Of Two Vectors

Find the dot product of . Note that both vectors are in polar form.

Explanation:

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:

### Example Question #7 : Find The Dot Product Of Two Vectors

Vectors  and .

Find the dot product

Explanation:

First rewrite the vectors in a bracket form

and

.

The dot product

### Example Question #8 : Find The Dot Product Of Two Vectors

Given vectors

and

Determine the dot product of vectors  and

Explanation:

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

### Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Tell whether the following two vectors are perpendicular or parallel and why.

Parallel, because their dot product is zero.

Perpendicular, because their dot product is one.

Neither perpendicular nor parallel, because their dot product is neither zero nor one.

Parallel, because their dot product is one.

Perpendicular, because their dot product is zero.

Perpendicular, because their dot product is zero.

Explanation:

Two vectors are perpendicular if their dot product is zero, and parallel if their dot product is 1.

Take the dot product of our two vectors to find the answer:

Using our given vectors:

Thus our two vectors are perpendicular.

### Example Question #1 : Determine If Two Vectors Are Parallel Or Perpendicular

Which of the following pairs of vectors are perpendicular?