Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

Create An Account

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #2 : Cross Product

For what angle(s) is the cross product $\vec{A} \times \vec{B}=0$ ?

Possible Answers:

$45 ^{\circ}$

$270^{\circ}$

$90^{\circ}$

$30^{\circ}$

$0 ^{\circ}, 180 ^{\circ}, 360 ^{\circ}$

Correct answer:

$0 ^{\circ}, 180 ^{\circ}, 360 ^{\circ}$

Explanation:

We have the following equation that relates the cross product of two vectors $\vec{A} \times \vec{B}$ to the relative angle between them $\theta$ , written as

$\frac{\vec{A} \times \vec{B}}{\left| \vec{A} \right| \left| \vec{B} \right|}= \sin \theta$ .

From this, we can see that the numerator, or cross product, will be whenever $\sin \theta=0$ . This will be true for all even multiples of $\frac{\pi}{2}$ . Therefore, we find that the cross product of two vectors will be for $\theta = 0(0^{\circ}), \pi (180^{\circ}), 2\pi (360^{\circ})$ .

Report an Error

Example Question #361 : Calculus 3

Evaluate $<1,0> \times <0,8>$

Possible Answers:

<0,0,0>

<0,8>

<0>

<0,0>

None of the other answers

Correct answer:

None of the other answers

Explanation:

It is not possible to take the cross product of -component vectors. The definition of the cross product states that the two vectors must each have components. So the above problem is impossible.

Report an Error

Example Question #7 : Cross Product

Compute $<1,0,4> \times <1,1,1>$ .

Possible Answers:

<3,4,-1>

<-3,-4,1>

<0,0,0>

<4,-3,-1>

<-4,3,1>

Correct answer:

<-4,3,1>

Explanation:

To evaluate the cross product, we use the determinant formula

$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1& a_2 & a_3 \\ b_1& b_2 & b_3 \end{vmatrix}$

So we have

$<1,0,4> \times <1,1,1>$

$= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1& 0 & 4 \\ 1& 1 & 1 \end{vmatrix}$

$= \begin{vmatrix} 0 & 4 \\ 1 & 1 \end{vmatrix} \mathbf{i} -\begin{vmatrix} 1 & 4 \\ 1& 1 \end{vmatrix} \mathbf{j} + \begin{vmatrix} 1 & 0 \\ 1 & 1 \end{vmatrix} \mathbf{k}$ . (Use cofactor expansion along the top row. This is typically done when taking any cross products)

$= -4\mathbf{i}-(-3)\mathbf{j}+1\mathbf{k}$

= <-4,3,1>

Report an Error

Example Question #2 : Cross Product

Evaluate $<1,1,0> \times <8,0,8>$ .

Possible Answers:

<-1,-1,1>

None of the other answers

<1,1,-1>

<8,-8,-8>

<-8,8,8>

Correct answer:

<8,-8,-8>

Explanation:

To evaluate the cross product, we use the determinant formula

$\mathbf{a} \times \mathbf{b} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ a_1& a_2 & a_3 \\ b_1& b_2 & b_3 \end{vmatrix}$

So we have

$<1,1,0> \times <8,0,8>$

$= \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1& 1 & 0 \\ 8& 0 & 8 \end{vmatrix}$

$= \begin{vmatrix} 1 & 0 \\ 0 & 8 \end{vmatrix} \mathbf{i} -\begin{vmatrix} 1 & 0 \\ 8& 8 \end{vmatrix} \mathbf{j} + \begin{vmatrix} 1 & 1 \\ 8 & 0 \end{vmatrix} \mathbf{k}$ . (Use cofactor expansion along the top row. This is typically done when taking any cross products)

$= 8\mathbf{i}-(8)\mathbf{j}+(-8)\mathbf{k}$

= <8,-8,-8>

Report an Error

Example Question #9 : Cross Product

Find the cross product of the two vectors.

u=<4,3,7>

v=<5,7,10>

Possible Answers:

<9,10,17>

<-19,-5,13>

<20,21,70>

<-1,-4,-3>

Correct answer:

<-19,-5,13>

Explanation:

To find the cross product, we solve for the determinant of the matrix

$\begin{vmatrix} i&j &k \\ 4&3 &7 \\ 5&7 &10 \end{vmatrix}$

The determinant equals

$=det(\begin{vmatrix} 3&7 \\ 7&10 \end{vmatrix})i-det(\begin{vmatrix} 4&7 \\ 5&10 \end{vmatrix})j+det(\begin{vmatrix} 4&3 \\ 5&7 \end{vmatrix})k$

=(3*10-7*7)i-(4*10-5*7)j+(4*7-5*3)k

=-19i-5j+13k

=<-19,-5,13>

As the cross-product.

Report an Error

Example Question #1 : Cross Product

Find the cross product of the two vectors.

u=<1,0,0>

v=<0,1,0>

Possible Answers:

<0,0,1>

<1,-1,0>

<1,1,0>

<0,0,0>

Correct answer:

<0,0,1>

Explanation:

To find the cross product, we solve for the determinant of the matrix

$\begin{vmatrix} i&j &k \\ 1&0 &0 \\ 0&1 &0 \end{vmatrix}$

The determinant equals

$=det(\begin{vmatrix} 0&0 \\ 1&0 \end{vmatrix})i-det(\begin{vmatrix} 1&0 \\ 0&0 \end{vmatrix})j+det(\begin{vmatrix} 1&0 \\ 0&1 \end{vmatrix})k$

=(0*0-1*0)i-(1*0-0*0)j+(1*1-0*0)k

=0i-0j+1k

=<0,0,1>

As the cross-product.

Report an Error

Example Question #11 : Cross Product

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=2\widehat{i}+4\widehat{k}$ and $\overrightarrow{b}=\widehat{i}+3\widehat{j}-2\widehat{k}$

Possible Answers:

$-12\widehat{i}-8\widehat{j}+6\widehat{k}$

$12\widehat{i}+8\widehat{j}+6\widehat{k}$

$-12\widehat{i}+8\widehat{j}-6\widehat{k}$

$-12\widehat{i}+8\widehat{j}+6\widehat{k}$

$8\widehat{i}-12\widehat{j}+6\widehat{k}$

Correct answer:

$-12\widehat{i}+8\widehat{j}+6\widehat{k}$

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=2\widehat{i}+4\widehat{k}$ and $\overrightarrow{b}=\widehat{i}+3\widehat{j}-2\widehat{k}$

$\overrightarrow{a}\times \overrightarrow{b}=(-12)\widehat{i}+(4+4)\widehat{j}+(6)\widehat{k}=-12\widehat{i}+8\widehat{j}+6\widehat{k}$

Report an Error

Example Question #12 : Cross Product

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=5\widehat{i}-5\widehat{j}$ and $\overrightarrow{b}=\widehat{i}+\widehat{j}$

Possible Answers:

$15\widehat{k}$

$-20\widehat{k}$

$10\widehat{k}$

$25\widehat{k}$

Correct answer:

$10\widehat{k}$

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=5\widehat{i}-5\widehat{j}$ and $\overrightarrow{b}=\widehat{i}+\widehat{j}$

$\overrightarrow{a}\times \overrightarrow{b}=(0+5+5+0)\widehat{k}=10\widehat{k}$

Report an Error

Example Question #12 : Cross Product

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=3\widehat{i}+9\widehat{j}$ and $\overrightarrow{b}=4\widehat{i}-16\widehat{j}$

Possible Answers:

$-84\widehat{k}$

$91\widehat{k}$

$-132\widehat{k}$

$-37\widehat{k}$

$156\widehat{k}$

Correct answer:

$-84\widehat{k}$

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=3\widehat{i}+9\widehat{j}$ and $\overrightarrow{b}=4\widehat{i}-16\widehat{j}$

$\overrightarrow{a}\times \overrightarrow{b}=(0-48-36+0)\widehat{k}=-84\widehat{k}$

Report an Error

Example Question #14 : Cross Product

Determine the cross product $\overrightarrow{a}\times\overrightarrow{b}$ , if $\overrightarrow{a}=-10\widehat{j}$ and $\overrightarrow{b}=3\widehat{i}+6\widehat{j}$

Possible Answers:

$-90\widehat{k}$

$45\widehat{k}$

$90\widehat{k}$

$-30\widehat{k}$

$30\widehat{k}$

Correct answer:

$30\widehat{k}$

Explanation:

The cross product of two vectors is a new vector. In order to find the cross product, it is useful to know the following relationships between the directional vectors:

$\widehat{i}\times\widehat{j}=\widehat{k};\widehat{j}\times\widehat{i}=-\widehat{k}$

$\widehat{j}\times\widehat{k}=\widehat{i};\widehat{k}\times\widehat{j}=-\widehat{i}$

$\widehat{k}\times\widehat{i}=\widehat{j};\widehat{i}\times\widehat{k}=-\widehat{j}$

$\widehat{i}\times\widehat{i}=\widehat{j}\times\widehat{j}=\widehat{k}\times\widehat{k}=0$

Note how the sign changes when terms are reordered; order matters!

Scalar values (the numerical coefficients) multiply through, e.g:

$r\widehat{i}\times s\widehat{j}=(rs)\widehat{k}$

With these principles in mind, we can calculate the cross product of our vectors $\overrightarrow{a}=-10\widehat{j}$ and $\overrightarrow{b}=3\widehat{i}+6\widehat{j}$

$\overrightarrow{a}\times \overrightarrow{b}=(30+0)\widehat{k}=30\widehat{k}$

Report an Error

View Calculus 3 Tutors

MrG
Certified Tutor

Long Island University-Brooklyn Campus, Master of Science, Applied Mathematics.

View Calculus 3 Tutors

Magdy
Certified Tutor

Cairo University, Egypt, Bachelor of Science, Electrical Engineering. New Mexico State University-Main Campus, Doctor of Phil...

View Calculus 3 Tutors

Dossevi
Certified Tutor

cole Ste Genevive Versailles, Bachelor of Science, Mathematics. Institut National Polytechnique de Grenoble, Master of Scienc...

All Calculus 3 Resources

6 Diagnostic Tests 373 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in Dallas Fort Worth, Calculus Tutors in Philadelphia, MCAT Tutors in Atlanta, Biology Tutors in Denver, Chemistry Tutors in San Diego, SAT Tutors in New York City, MCAT Tutors in Miami, Algebra Tutors in Chicago, Algebra Tutors in Dallas Fort Worth, MCAT Tutors in Seattle

Popular Courses & Classes

SSAT Courses & Classes in Phoenix, SAT Courses & Classes in Dallas Fort Worth, SSAT Courses & Classes in San Diego, MCAT Courses & Classes in Seattle, SAT Courses & Classes in Atlanta, LSAT Courses & Classes in Miami, SSAT Courses & Classes in Denver, GRE Courses & Classes in San Francisco-Bay Area, ACT Courses & Classes in Seattle, ACT Courses & Classes in San Francisco-Bay Area

Popular Test Prep

SAT Test Prep in New York City, ACT Test Prep in San Diego, GMAT Test Prep in Houston, SSAT Test Prep in Dallas Fort Worth, MCAT Test Prep in Dallas Fort Worth, SSAT Test Prep in Denver, SAT Test Prep in Chicago, GMAT Test Prep in San Diego, LSAT Test Prep in Philadelphia, ACT Test Prep in Seattle

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Calculus 3 : Calculus 3

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #2 : Cross Product

Example Question #361 : Calculus 3

Example Question #7 : Cross Product

Example Question #2 : Cross Product

Example Question #9 : Cross Product

Example Question #1 : Cross Product

Example Question #11 : Cross Product

Example Question #12 : Cross Product

Example Question #12 : Cross Product

Example Question #14 : Cross Product

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: