Precalculus : Matrices and Vectors

Study concepts, example questions & explanations for Precalculus

varsity tutors app store varsity tutors android store

Example Questions

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

Find the dot product of the two vectors

and

.

 

 
Possible Answers:

Correct answer:

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:

Possible Answers:

Correct answer:

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

.

Possible Answers:

Correct answer:

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

.

Possible Answers:

Correct answer:

Explanation:

Let

The dot product  is equal to

.

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

Determine the dot product of  and .

Possible Answers:

Correct answer:

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.

Possible Answers:

Correct answer:

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, :

Srt 30 60 90

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:

Srt 45 45 90

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 

Possible Answers:

Correct answer:

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 

Possible Answers:

Correct answer:

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.

Possible Answers:

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.

Correct answer:

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?

Possible Answers:

Correct answer:

Explanation:

Two vectors are perpendicular when their dot product equals to .

Recall how to find the dot product of two vectors  and 

The correct choice is 

 

Learning Tools by Varsity Tutors