Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Normal Vectors

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 #91 : Normal Vectors

Find the normal vector to the plane containing the vectors $\left \langle0,-5,-4 \right \rangle$ and $\left \langle2,-4,7\right \rangle$

Possible Answers:

$\left \langle -51,-5,10\right \rangle$

$\left \langle -41,-8,10\right \rangle$

$\left \langle -50,-8,9\right \rangle$

$\left \langle -51,-8,10\right \rangle$

Correct answer:

$\left \langle -51,-8,10\right \rangle$

Explanation:

To find the normal vector to a plane containing vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you take the cross product of the two vectors.

To find the cross product between the two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you 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

$determinant=\hat{i}(-35-16)-\hat{j}(0+8)+\hat{k}(0+10)=\left \langle -51,-8,10\right \rangle$

Report an Error

Example Question #92 : Normal Vectors

Find the normal vector to the plane containing the vectors $\left \langle -2,0,6\right \rangle$ and $\left \langle 2,-3,-1\right \rangle$

Possible Answers:

$\left \langle 18,11,5\right \rangle$

$\left \langle 10,13,6\right \rangle$

$\left \langle 18,-14,6\right \rangle$

$\left \langle 18,14,6\right \rangle$

Correct answer:

$\left \langle 18,14,6\right \rangle$

Explanation:

To obtain the normal vector to a plane containing two vectors $a=\left \langle a_1,a_2,a_3\right \rangle$ and $b=\left \langle b_1,b_2,b_3\right \rangle$ , you compute 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 this formula, we evaluate using the vectors from the problem statement:

$\hat{i}(0+18)-\hat{j}(-2-12)+\hat{k}(6-0)=\left \langle 18,14,6\right \rangle$

Report an Error

Example Question #93 : Normal Vectors

Find the normal vector to the plane that contains the vectors $\left \langle 0,4,9\right \rangle$ and $\left \langle 0,7,5\right \rangle$

Possible Answers:

$\left \langle -27,3,6\right \rangle$

$\left \langle -25,0,0\right \rangle$

$\left \langle -27,0,0\right \rangle$

$\left \langle -27,1,1\right \rangle$

Correct answer:

$\left \langle -27,0,0\right \rangle$

Explanation:

$determinant=\hat{i}(a_2b_3-b_2a_3)-\hat{j}(a_1b_3-b_1a_3)+\hat{k}(a_1b_2-b_1a_2)$

Using this formula, we evaluate using the vectors from the problem statement:

$\hat{i}(36-63)-\hat{j}(0-0)+\hat{k}(0-0)=\left \langle -27,0,0\right \rangle$

Report an Error

Example Question #94 : Normal Vectors

Find the normal vector to the plane that contains the vectors $\left \langle 5,0,-6\right \rangle$ and $\left \langle 7,0,2\right \rangle$

Possible Answers:

$\left \langle -3,16,3\right \rangle$

$\left \langle 0,51,0\right \rangle$

$\left \langle -3,12,0\right \rangle$

$\left \langle 0,52,0\right \rangle$

Correct answer:

$\left \langle 0,52,0\right \rangle$

Explanation:

$determinant=\hat{i}(a_2b_3-b_2a_3)-\hat{j}(a_1b_3-b_1a_3)+\hat{k}(a_1b_2-b_1a_2)$

Using this formula, we evaluate using the vectors from the problem statement:

$\hat{i}(0-0)-\hat{j}(10+42)+\hat{k}(0-0)=\left \langle 0,52,0\right \rangle$

Report an Error

Example Question #2591 : Calculus 3

Find the normal vector to the plane that contains the vectors $\left \langle 2,0,9\right \rangle$ and $\left \langle -3,0,12\right \rangle$

Possible Answers:

$\left \langle 0,51,0\right \rangle$

$\left \langle 27,33,3\right \rangle$

$\left \langle -2,48,0\right \rangle$

$\left \langle 3,5,18\right \rangle$

Correct answer:

$\left \langle 0,51,0\right \rangle$

Explanation:

$determinant=\hat{i}(a_2b_3-b_2a_3)-\hat{j}(a_1b_3-b_1a_3)+\hat{k}(a_1b_2-b_1a_2)$

Using this formula, we evaluate using the vectors from the problem statement:

$\hat{i}(0-0)-\hat{j}(24+27)+\hat{k}(0-0)=\left \langle 0,51,0\right \rangle$

Report an Error

Example Question #2591 : Calculus 3

Find the Unit Normal Vector to the given plane.

3x+3y+4z=10 .

Possible Answers:

V_n=(3, 3, 4 )

$V_n=(\frac{1}{\sqrt{34}}, \frac{1}{\sqrt{34}}, \frac{1}{\sqrt{34}} )$

$V_n=(\frac{3}{\sqrt{34}}, \frac{3}{\sqrt{34}}, \frac{4}{\sqrt{34}} )$

$V_n=(\frac{3}{{34}}, \frac{3}{{34}}, \frac{4}{{34}} )$

$V_n=(\frac{1}{{34}}, \frac{1}{{34}}, \frac{1}{{34}} )$

Correct answer:

$V_n=(\frac{3}{\sqrt{34}}, \frac{3}{\sqrt{34}}, \frac{4}{\sqrt{34}} )$

Explanation:

Recall the definition of the Unit Normal Vector.

$V_n=\frac{V}{||V||}$

Let V=(3,3,4)

$||V||=\sqrt{3^2+3^2+4^2}=\sqrt{34}$

$V_n=(\frac{3}{\sqrt{34}}, \frac{3}{\sqrt{34}}, \frac{4}{\sqrt{34}} )$

Report an Error

Example Question #2591 : Calculus 3

Find the Unit Normal Vector to the given plane.

3x+3y+4z=10 .

Possible Answers:

$V_n=(\frac{3}{{34}}, \frac{3}{{34}}, \frac{4}{{34}} )$

$V_n=(\frac{1}{{34}}, \frac{1}{{34}}, \frac{1}{{34}} )$

V_n=(3, 3, 4 )

$V_n=(\frac{3}{\sqrt{34}}, \frac{3}{\sqrt{34}}, \frac{4}{\sqrt{34}} )$

$V_n=(\frac{1}{\sqrt{34}}, \frac{1}{\sqrt{34}}, \frac{1}{\sqrt{34}} )$

Correct answer:

$V_n=(\frac{3}{\sqrt{34}}, \frac{3}{\sqrt{34}}, \frac{4}{\sqrt{34}} )$

Explanation:

Recall the definition of the Unit Normal Vector.

$V_n=\frac{V}{||V||}$

Let V=(3,3,4)

$||V||=\sqrt{3^2+3^2+4^2}=\sqrt{34}$

$V_n=(\frac{3}{\sqrt{34}}, \frac{3}{\sqrt{34}}, \frac{4}{\sqrt{34}} )$

Report an Error

View Calculus 3 Tutors

Philipp
Certified Tutor

University of Nevada-Las Vegas, Bachelor of Science, Mechanical Engineering.

View Calculus 3 Tutors

Peter
Certified Tutor

Kansas State University, Bachelors, Mathematics. Bowling Green State University-Main Campus, Master of Science, Management In...

View Calculus 3 Tutors

Reuben
Certified Tutor

University of Alberta, Bachelor of Science, Mathematics.

All Calculus 3 Resources

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

Popular Subjects

SSAT Tutors in New York City, SAT Tutors in San Francisco-Bay Area, GMAT Tutors in Los Angeles, SAT Tutors in Miami, Physics Tutors in Los Angeles, ACT Tutors in San Francisco-Bay Area, MCAT Tutors in Boston, Physics Tutors in Dallas Fort Worth, Calculus Tutors in Phoenix, Biology Tutors in San Francisco-Bay Area

Popular Courses & Classes

GMAT Courses & Classes in Washington DC, SAT Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in San Diego, MCAT Courses & Classes in Washington DC, ISEE Courses & Classes in Philadelphia, Spanish Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Seattle, ACT Courses & Classes in Boston, GRE Courses & Classes in Houston, ISEE Courses & Classes in New York City

Popular Test Prep

SSAT Test Prep in Dallas Fort Worth, SAT Test Prep in Seattle, SSAT Test Prep in Houston, GMAT Test Prep in Washington DC, LSAT Test Prep in Boston, SSAT Test Prep in Chicago, GMAT Test Prep in Houston, GRE Test Prep in San Diego, ACT Test Prep in Phoenix, ISEE 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 : Normal Vectors

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #91 : Normal Vectors

Example Question #92 : Normal Vectors

Example Question #93 : Normal Vectors

Example Question #94 : Normal Vectors

Example Question #2591 : Calculus 3

Example Question #2591 : Calculus 3

Example Question #2591 : Calculus 3

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: