Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Cross Product

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 #177 : Vectors And Vector Operations

Find the cross product, in vector form, of the two vectors:

$\left \langle 3, 2, 3\right \rangle, \left \langle 10, 2, 1\right \rangle$

Possible Answers:

$\left \langle 4, 27, 14\right \rangle$

$\left \langle 0, 0, 0\right \rangle$

$\left \langle -4, 27, -14\right \rangle$

$\left \langle 8, 33, 26\right \rangle$

Correct answer:

$\left \langle -4, 27, -14\right \rangle$

Explanation:

First, we must write the determinant in order to take the cross product of the two vectors:

$\begin{vmatrix} \hat{i}& \hat{j}&\hat{k} \\ 3&2 &3 \\ 10&2 &1 \end{vmatrix}$

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

$2\hat{i}+6\hat{k}+30\hat{j}-(20\hat{k}+6\hat{i}+3\hat{j})=-4\hat{i}+27\hat{j}-14\hat{k=\left \langle -4, 27, -14\right \rangle}$

Report an Error

Example Question #178 : Vectors And Vector Operations

Find the cross product, in vector form, of the two vectors:

$\left \langle 3, 5x, 2\right \rangle, \left \langle 2y, 3x, z\right \rangle$

Possible Answers:

$\left \langle 5xz-6x, 4y-9, 9x-10xy\right \rangle$

$\left \langle 5xz+6x, 4y+9, 9x+10xy\right \rangle$

$\left \langle 5xz-6x, 4y-9, -x\right \rangle$

$\left \langle -5xz+6x, -4y+9, -9x+10xy\right \rangle$

Correct answer:

$\left \langle 5xz-6x, 4y-9, 9x-10xy\right \rangle$

Explanation:

First, we must write the determinant in order to take the cross product of the two vectors:

$\begin{vmatrix} \hat{i}& \hat{j}&\hat{k} \\ 3&5x &2 \\ 2y&3x &z \end{vmatrix}$

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

$5xz\hat{i}+9x\hat{k}+4y\hat{j}-(10xy\hat{k}+6x\hat{i}+9\hat{j})=(5xz-6)\hat{i}+(4y-9)\hat{j}+(9x-10xy)\hat{k=\left \langle 5xz-6x, 4y-9, 9x-10xy\right \rangle}$

Report an Error

Example Question #179 : Vectors And Vector Operations

Find the cross product of the following vectors, in vector form:

$\left \langle a, b, c\right \rangle, \left \langle x\cos(x), \sin(z), xy\right \rangle$

Possible Answers:

$\left \langle bxy+c\sin(z), cx\cos(x)-axy, a\sin(x)+c\sin(z)\right \rangle$

$\left \langle bxy+c\sin(z), cx\cos(x)+axy, a\sin(x)+c\sin(z)\right \rangle$

$\left \langle -bxy+c\sin(z), -cx\cos(x)+axy, -a\sin(x)+c\sin(z)\right \rangle$

$\left \langle bxy-c\sin(z), cx\cos(x)-axy, a\sin(x)-c\sin(z)\right \rangle$

Correct answer:

$\left \langle bxy-c\sin(z), cx\cos(x)-axy, a\sin(x)-c\sin(z)\right \rangle$

Explanation:

First, we must write the determinant in order to take the cross product of the two vectors:

$\begin{vmatrix} \hat{i}& \hat{j}&\hat{k} \\ a&b &c\\ x\cos(x)&\sin(z) &xy \end{vmatrix}$

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

Next, we take the cross product. One can do this by multiplying across from the top left to the lower right, and continuing downward, and then subtracting the terms multiplied from top right to the bottom left:
$bxy\hat{i}+a\sin(z)\hat{k}+cx\cos(x)\hat{j}-(bx\cos(x)\hat{k}+c\sin(z)\hat{i}+axy\hat{j})=(bxy-c\sin(z))\hat{i}+(cx\cos(x)-axy)\hat{j}+(a\sin(x)-c\sin(z))\hat{k}=\left \langle bxy-c\sin(z), cx\cos(x)-axy, a\sin(x)-c\sin(z)\right \rangle$

Report an Error

Example Question #180 : Vectors And Vector Operations

Find the cross product between the vectors $\left \langle 3,1,7\right \rangle$ and $\left \langle 0,5,1\right \rangle$

Possible Answers:

$\left \langle -34,-3,15\right \rangle$

$\left \langle -30,-3,15\right \rangle$

$\left \langle -34,3,15\right \rangle$

$\left \langle -35,-3,10\right \rangle$

Correct answer:

$\left \langle -34,-3,15\right \rangle$

Explanation:

To find the cross product between the two vectors $\left \langle x_1,y_1,z_1\right \rangle$ and $\left \langle x_2,y_2,z_2\right \rangle$ , we find the determinant of the 3x3 matrix $\begin{bmatrix} i&j &k \\ x_1&y_1 & z_1\\ x_2&y_2 &z_2 \end{bmatrix}$

Plugging in the vectors and solving, we get

$i(1-35)-j(3-0)+k(15-0)=-34i-3j+15k=\left \langle-34,-3,15 \right \rangle$

Report an Error

Example Question #181 : Vectors And Vector Operations

Find the cross product between the vectors $\left \langle 2,-5,7\right \rangle$ and $\left \langle 0,-3,1\right \rangle$

Possible Answers:

$\left \langle 16,-2,-6\right \rangle$

$\left \langle 16,-1,-6\right \rangle$

$\left \langle 16,2,-6\right \rangle$

$\left \langle 15,-2,-6\right \rangle$

Correct answer:

$\left \langle 16,-2,-6\right \rangle$

Explanation:

To find the cross product between two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we find the determinant of the 3x3 matrix $\begin{bmatrix} i&j &k \\ a_1&a_2 &a_3 \\ b_1&b_2 &b_3 \end{bmatrix}$ which follows the formula determinant= i(a_2b_3-b_2a_3)-j(a_1b_3-b_1a_3)+k(a_1b_2-b_1a_2)

Applying to the vectors from the problem statement, we get

$i(-5+21)-j(2-0)+k(-6-0)=\left \langle 16,-2,-6\right \rangle$

Report an Error

Example Question #431 : Calculus 3

Find the cross product between the vectors $\left \langle 1,-4,6\right \rangle$ and $\left \langle -1,0,5\right \rangle$

Possible Answers:

$\left \langle -20,11,-4\right \rangle$

$\left \langle -22,-11,-4\right \rangle$

$\left \langle -20,-10,-6\right \rangle$

$\left \langle -20,-11,-4\right \rangle$

Correct answer:

$\left \langle -20,-11,-4\right \rangle$

Explanation:

Applying to the vectors from the problem statement, we get

$i(-20-0)-j(5+6)+k(0-4)=\left \langle -20,-11,-4\right \rangle$

Report an Error

Example Question #71 : Cross Product

Find the cross product between the two vectors:

$\left \langle 4, 9, 1\right \rangle, \left \langle 3, -1, -1\right \rangle$

Possible Answers:

$\left \langle -8, 7, -26\right \rangle$

$\left \langle -9, -4, 3\right \rangle$

$\left \langle -10, -1, 18\right \rangle$

$\left \langle 8, 7, 18\right \rangle$

Correct answer:

$\left \langle -8, 7, -26\right \rangle$

Explanation:

First, we can write the determinant in order to take the cross product of the two vectors:

$\begin{vmatrix} \hat{i}& \hat{j}&\hat{k} \\ 4& 9 &1 \\ 3& -1&-1 \end{vmatrix}$

where i, j, and k are the unit vectors corresponding to the x, y, and z direction respectively.

$-9\hat{i}+-4\hat{k}+3\hat{j}-(22\hat{k}-\hat{i}-4\hat{j})= -8\hat{i}+7\hat{j}+18\hat{k}=\left \langle -8, 7, 18\right \rangle$

Report an Error

Example Question #432 : Calculus 3

Find the cross product between the vectors $\left \langle -3,-4,0\right \rangle$ and $\left \langle -1,1,4\right \rangle$

Possible Answers:

$\left \langle -16,10,-7\right \rangle$

$\left \langle -15,12,-6\right \rangle$

$\left \langle -16,-12,-7\right \rangle$

$\left \langle -16,12,-7\right \rangle$

Correct answer:

$\left \langle -16,12,-7\right \rangle$

Explanation:

To find the cross product between two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we take the determinant of the 3x3 matrix

$\begin{bmatrix} \hat{i}&\hat{j} &\hat{k} \\ a_1&a_2 &a_3 \\ b_1& b_2&b_3 \end{bmatrix}$ . $determinant=\hat{i}(a_2b_3-b_2a_3)-\hat{j}(a_1b_3-b_1a_3)+\hat{k}(a_1b_2-b_1a_2)$

Using the vectors from the problem statement, we get

$determinant=\hat{i}(-16-0)-\hat{j}(-12-0)+\hat{k}(-3-4)=\left \langle -16,12,-7\right \rangle$

Report an Error

Example Question #433 : Calculus 3

Find the cross product between the vectors $\left \langle 0,1,7\right \rangle$ and $\left \langle 5,1,-2\right \rangle$

Possible Answers:

$\left \langle 9,35,-5\right \rangle$

$\left \langle 9,35,-3\right \rangle$

$\left \langle 8,25,-5\right \rangle$

$\left \langle 9,-35,-5\right \rangle$

Correct answer:

$\left \langle 9,35,-5\right \rangle$

Explanation:

To find the cross product between two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we take the determinant of the 3x3 matrix

$\begin{bmatrix} \hat{i}&\hat{j} &\hat{k} \\ a_1&a_2 &a_3 \\ b_1& b_2&b_3 \end{bmatrix}$ . $determinant=\hat{i}(a_2b_3-b_2a_3)-\hat{j}(a_1b_3-b_1a_3)+\hat{k}(a_1b_2-b_1a_2)$

Using the vectors from the problem statement, we get

$determinant=\hat{i}(2+7)-\hat{j}(0-35)+\hat{k}(0-5)=\left \langle 9,35,-5\right \rangle$

Report an Error

Example Question #76 : Cross Product

Find the cross product between the vectors $\left \langle 0,8,4\right \rangle$ and $\left \langle 1,4,-5\right \rangle$

Possible Answers:

$\left \langle -61,-4,-8\right \rangle$

$\left \langle -51,4,-4\right \rangle$

$\left \langle -61,4,-8\right \rangle$

$\left \langle -60,5,-9\right \rangle$

Correct answer:

$\left \langle -61,4,-8\right \rangle$

Explanation:

To find the cross product between two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , we take the determinant of the 3x3 matrix

$\begin{bmatrix} \hat{i}&\hat{j} &\hat{k} \\ a_1&a_2 &a_3 \\ b_1& b_2&b_3 \end{bmatrix}$ . $determinant=\hat{i}(a_2b_3-b_2a_3)-\hat{j}(a_1b_3-b_1a_3)+\hat{k}(a_1b_2-b_1a_2)$

Using the vectors from the problem statement, we get

$determinant=\hat{i}(-45-16)-\hat{j}(0-4)+\hat{k}(0-8)=\left \langle -61,4,-8\right \rangle$

Report an Error

View Calculus 3 Tutors

Gregory
Certified Tutor

Pennsylvania State University-Main Campus, Bachelor of Science, Chemical Engineering. Carnegie Mellon University, Doctor of P...

View Calculus 3 Tutors

Muhammad
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Mechanical Engineering.

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

GMAT Tutors in Seattle, MCAT Tutors in Los Angeles, LSAT Tutors in Chicago, MCAT Tutors in New York City, Spanish Tutors in Dallas Fort Worth, GMAT Tutors in Phoenix, Statistics Tutors in Phoenix, Statistics Tutors in Houston, English Tutors in Atlanta, Statistics Tutors in San Francisco-Bay Area

Popular Courses & Classes

ISEE Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in Chicago, Spanish Courses & Classes in Chicago, SSAT Courses & Classes in Phoenix, ACT Courses & Classes in Chicago, SSAT Courses & Classes in Washington DC, MCAT Courses & Classes in New York City, MCAT Courses & Classes in Atlanta, SAT Courses & Classes in San Diego, SAT Courses & Classes in Washington DC

Popular Test Prep

GRE Test Prep in Miami, ACT Test Prep in San Francisco-Bay Area, MCAT Test Prep in San Diego, ISEE Test Prep in Atlanta, SAT Test Prep in Boston, SSAT Test Prep in San Diego, MCAT Test Prep in Atlanta, MCAT Test Prep in Dallas Fort Worth, ACT Test Prep in Houston, SSAT Test Prep in Chicago

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 : Cross Product

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #177 : Vectors And Vector Operations

Example Question #178 : Vectors And Vector Operations

Example Question #179 : Vectors And Vector Operations

Example Question #180 : Vectors And Vector Operations

Example Question #181 : Vectors And Vector Operations

Example Question #431 : Calculus 3

Example Question #71 : Cross Product

Example Question #432 : Calculus 3

Example Question #433 : Calculus 3

Example Question #76 : Cross Product

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: