Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Parametric, Polar, and Vector

Study concepts, example questions & explanations for Calculus 2

Create An Account

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #205 : Vector

What is the norm of $a=\left \langle 5,-5,1\right \rangle$ ?

Possible Answers:

$\sqrt{51}$

$5\sqrt{2}$

$4\sqrt{3}$

$2\sqrt{13}$

Correct answer:

$\sqrt{51}$

Explanation:

In order to find the norm of a vector, we must first find the sum of the squares of the vector's individual elements and then take the square root of that sum. Given $a=\left \langle 5,-5,1\right \rangle$ ,

$\left \| a\right \|=\sqrt{(5)^{2}+(-5)^{2}+(1)^{2}}$

$\left \| a\right \|=\sqrt{25+25+1}$

$\left \| a\right \|=\sqrt{50+1}$

$\left \| a\right \|=\sqrt{51}$

Report an Error

Example Question #47 : Vector Calculations

Find the cross product of $a=\left \langle 2,3,4\right \rangle$ and $b=\left \langle 3,2,1\right \rangle$ .

Possible Answers:

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

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

None of the above

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

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

Correct answer:

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

Explanation:

In order to find the cross product of two three-dimensional vectors, we must find the determinant of a matrix comprised of the vectors' elements. That is, if $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ and $b=\left \langle b_{1},b_{2},b_{3}\right \rangle$ , then

$a\times b=\begin{Bmatrix} i&j &k \\ a_{1}&a_{2} &a_{3} \\ b_{1}&b_{2} &b_{3} \end{Bmatrix}=\left \langle a_{2}b_{3}-a_{3}b_{2}, a_{1}b_{3}-a_{3}b_{1},a_{1}b_{2}-a_{2}b_{1} \right \rangle$ .

Given $a=\left \langle 2,3,4\right \rangle$ and $b=\left \langle 3,2,1\right \rangle$ , the cross product $a \times b$ is:

$\begin{Bmatrix} i&j &k \\ 2&3 &4 \\ 3&2 &1 \end{Bmatrix}$

$=\left \langle (3)(1)-(4)(2), (2)(1)-(4)(3),(2)(2)-(3)(3) \right \rangle$

$=\left \langle 3-8,2-12, 4-9\right \rangle$

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

Report an Error

Example Question #206 : Vector

Find the cross product of $a=\left \langle 1,0,-1\right \rangle$ and $b=\left \langle 0,-1,1\right \rangle$ .

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

In order to find the cross product of two vectors, we must find the determinant of a matrix comprised of the vectors' elements. That is, if $a=\left \langle a_{1},a_{2},a_{3}\right \rangle$ and $b=\left \langle b_{1},b_{2},b_{3}\right \rangle$ , then

$a\times b=\begin{Bmatrix} i&j &k \\ a_{1}&a_{2} &a_{3} \\ b_{1}&b_{2} &b_{3} \end{Bmatrix}=\left \langle a_{2}b_{3}-a_{3}b_{2}, a_{1}b_{3}-a_{3}b_{1},a_{1}b_{2}-a_{2}b_{1} \right \rangle$ .

Given $a=\left \langle 1,0,-1\right \rangle$ and $b=\left \langle 0,-1,1\right \rangle$ . the cross product $a \times b$ is:

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

$=\left \langle (0)(1)-(-1)(-1), (1)(1)-(-1)(0), (1)(-1)-(0)(0) \right \rangle$

$=\left \langle 0-1, 1-0,-1-0 \right \rangle$

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

Report an Error

Example Question #207 : Vector

What is the cross product of $a=\left \langle 7,1,-1\right \rangle$ and $b=\left \langle 1,-1, -7 \right \rangle$ ?

Possible Answers:

$\left \langle8, 48, -8 \right \rangle$

$\left \langle8, 48, 8 \right \rangle$

$\left \langle-8, -48, -8 \right \rangle$

$\left \langle8, -48, -8 \right \rangle$

Correct answer:

$\left \langle-8, -48, -8 \right \rangle$

Explanation:

$a\times b=\begin{Bmatrix} i&j &k \\ a_{1}&a_{2} &a_{3} \\ b_{1}&b_{2} &b_{3} \end{Bmatrix}=\left \langle a_{2}b_{3}-a_{3}b_{2}, a_{1}b_{3}-a_{3}b_{1},a_{1}b_{2}-a_{2}b_{1} \right \rangle$ .

Given $a=\left \langle 7,1,-1\right \rangle$ and $b=\left \langle 1,-1, -7 \right \rangle$ the cross product $a \times b$ is:

$\begin{Bmatrix} i&j &k \\ 7&1 &-1 \\ 1&-1 &-7 \end{Bmatrix}$

$=\left \langle (1)(-7)-(-1)(-1), (7)(-7)-(-1)(1), (7)(-1)-(1)(1) \right \rangle$

$=\left \langle(-7)-1, (-49)-(-1), (-7)-1 \right \rangle$

$=\left \langle-8, -48, -8 \right \rangle$

Report an Error

Example Question #208 : Vector

What is the cross product of $a=\left \langle 5,6,-2\right \rangle$ and $b=\left \langle -3,1, 9 \right \rangle$ ?

Possible Answers:

$\left \langle 56, 39, 23 \right \rangle$

$\left \langle -56, -39, -23 \right \rangle$

$\left \langle -56, 39, -23 \right \rangle$

$\left \langle -56, 39, 23 \right \rangle$

$\left \langle -56, -39, 23 \right \rangle$

Correct answer:

$\left \langle 56, 39, 23 \right \rangle$

Explanation:

$a\times b=\begin{Bmatrix} i&j &k \\ a_{1}&a_{2} &a_{3} \\ b_{1}&b_{2} &b_{3} \end{Bmatrix}=\left \langle a_{2}b_{3}-a_{3}b_{2}, a_{1}b_{3}-a_{3}b_{1},a_{1}b_{2}-a_{2}b_{1} \right \rangle$ .

Given $a=\left \langle 5,6,-2\right \rangle$ and $b=\left \langle -3,1, 9 \right \rangle$ , the cross product $a \times b$ is:

$\begin{Bmatrix} i&j &k \\ 5&6 &-2 \\ -3&1 &9 \end{Bmatrix}$

$=\left \langle (6)(9)-(-2)(1), (5)(9)-(-2)(-3), (5)(1)-(6)(-3) \right \rangle$

$=\left \langle 54+2, 45-6, 5+18 \right \rangle$

$=\left \langle 56, 39, 23 \right \rangle$

Report an Error

Example Question #51 : Vector Calculations

What is the cross product of $a=\left \langle -1,0,8\right \rangle$ and $b=\left \langle -8,-1, 0 \right \rangle$ ?

Possible Answers:

$\left \langle -8,-64, 1 \right \rangle$

$\left \langle -8,64, 1 \right \rangle$

$\left \langle 8,64, 1 \right \rangle$

$\left \langle -8,64, -1 \right \rangle$

$\left \langle -8,-64, -1 \right \rangle$

Correct answer:

$\left \langle 8,64, 1 \right \rangle$

Explanation:

$a\times b=\begin{Bmatrix} i&j &k \\ a_{1}&a_{2} &a_{3} \\ b_{1}&b_{2} &b_{3} \end{Bmatrix}=\left \langle a_{2}b_{3}-a_{3}b_{2}, a_{1}b_{3}-a_{3}b_{1},a_{1}b_{2}-a_{2}b_{1} \right \rangle$ .

Given $a=\left \langle -1,0,8\right \rangle$ and $b=\left \langle -8,-1, 0 \right \rangle$ , the cross product $a \times b$ is:

$\begin{Bmatrix} i&j &k \\ -1&0 &8 \\ -8&-1 &0 \end{Bmatrix}$

$=\left \langle (0)(0)-(8)(-1), (-1)(0)-(8)(-8), (-1)(-1)-(0)(-8) \right \rangle$

$=\left \langle 0+8, 0+64, 1-0 \right \rangle$

$=\left \langle 8,64, 1 \right \rangle$

Report an Error

Example Question #211 : Vector

What is the norm of $a=\left \langle 2,4,6\right \rangle$ ?

Possible Answers:

$2\sqrt{15}$

$2\sqrt{14}$

$3\sqrt{6}$

$3\sqrt{7}$

$\sqrt{55}$

Correct answer:

$2\sqrt{14}$

Explanation:

In order to find the norm of a vector, we must first find the sum of the squares of the vector's elements and take the square root of that sum. Given $a=\left \langle 2,4,6\right \rangle$ , then:

$\left \| a\right \|=\sqrt{(2)^{2}+(4)^{2}+(6)^{2}}$

$\left \| a\right \|=\sqrt{4+16+36}$

$\left \| a\right \|=\sqrt{20+36}$

$\left \| a\right \|=\sqrt{56}$

$\left \| a\right \|=\sqrt{4\times 14}$

$\left \| a\right \|=2\sqrt{14}$

Report an Error

Example Question #212 : Vector

What is the norm of $a=\left \langle 3,5,7\right \rangle$ ?

Possible Answers:

$3\sqrt{11}$

$3\sqrt{10}$

$\sqrt{83}$

Correct answer:

$\sqrt{83}$

Explanation:

In order to find the norm of a vector, we must first find the sum of the squares of the vector's elements and take the square root of that sum. Given $a=\left \langle 3,5,7\right \rangle$ , then:

$\left \| a\right \|=\sqrt{(3)^{2}+(5)^{2}+(7)^{2}}$

$\left \| a\right \|=\sqrt{9+25+49}$

$\left \| a\right \|=\sqrt{34+49}$

$\left \| a\right \|=\sqrt{83}$

Report an Error

Example Question #213 : Vector

What is the norm of $a=\left \langle9,3,6\right \rangle$ ?

Possible Answers:

$3\sqrt{13}$

$3\sqrt{11}$

$3\sqrt{10}$

$3\sqrt{14}$

Correct answer:

$3\sqrt{14}$

Explanation:

In order to find the norm of a vector, we must first find the sum of the squares of the vector's elements and take the square root of that sum. Given $a=\left \langle9,3,6\right \rangle$ , then:

$\left \| a\right \|=\sqrt{(9)^{2}+(3)^{2}+(6)^{2}}$

$\left \| a\right \|=\sqrt{81+9+36}$

$\left \| a\right \|=\sqrt{90+36}$

$\left \| a\right \|=\sqrt{126}$

$\left \| a\right \|=\sqrt{9\times 14}$

$\left \| a\right \|=3\sqrt{14}$

Report an Error

Example Question #1073 : Calculus Ii

Find the cross product of $a=\left \langle 5,7,2\right \rangle$ and $b=\left \langle 4,4,8\right \rangle$ .

Possible Answers:

None of the above

$\left \langle 48, 32, -8\right \rangle$

$\left \langle 20, 32, 8\right \rangle$

$\left \langle -20, -32, -8\right \rangle$

$\left \langle 20, -32, -8\right \rangle$

Correct answer:

$\left \langle 48, 32, -8\right \rangle$

Explanation:

$a\times b=\begin{Bmatrix} i&j &k \\ a_{1}&a_{2} &a_{3} \\ b_{1}&b_{2} &b_{3} \end{Bmatrix}=\left \langle a_{2}b_{3}-a_{3}b_{2}, a_{1}b_{3}-a_{3}b_{1},a_{1}b_{2}-a_{2}b_{1} \right \rangle$ .

Given $a=\left \langle 5,7,2\right \rangle$ and $b=\left \langle 4,4,8\right \rangle$ . the cross product $a \times b$ is:

$\begin{Bmatrix} i&j &k \\ 5&7 &2 \\ 4&4 &8 \end{Bmatrix}$

$=\left \langle (7)(8)-(2)(4), (5)(8)-(2)(4), (5)(4)-(7)(4) \right \rangle$

$=\left \langle 56-8, 40-8, 20-28\right \rangle$

$=\left \langle 48, 32, -8\right \rangle$

Report an Error

View Calculus 2 Tutors

Mohammed
Certified Tutor

University of Maryland-College Park, Bachelor of Science, Electrical Engineering. University of Maryland-College Park, Master...

View Calculus 2 Tutors

Pankaj
Certified Tutor

University of Delhi, Electrical Engineer, Electrical Engineering. Columbia University in the City of New York, Master of Arts...

View Calculus 2 Tutors

Robert
Certified Tutor

University of Guelph, Bachelor, Biochemistry.

All Calculus 2 Resources

9 Diagnostic Tests 308 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Spanish Tutors in Houston, Calculus Tutors in Washington DC, GMAT Tutors in San Francisco-Bay Area, Chemistry Tutors in Miami, Statistics Tutors in Miami, Computer Science Tutors in Phoenix, Reading Tutors in Houston, Statistics Tutors in Houston, SAT Tutors in Phoenix, French Tutors in Phoenix

Popular Courses & Classes

ACT Courses & Classes in Atlanta, ACT Courses & Classes in San Diego, SAT Courses & Classes in Miami, Spanish Courses & Classes in Los Angeles, SAT Courses & Classes in New York City, ISEE Courses & Classes in San Diego, SSAT Courses & Classes in San Diego, MCAT Courses & Classes in Los Angeles, LSAT Courses & Classes in San Diego, ACT Courses & Classes in Chicago

Popular Test Prep

MCAT Test Prep in Miami, ACT Test Prep in Atlanta, SAT Test Prep in Seattle, ISEE Test Prep in Philadelphia, SAT Test Prep in Houston, LSAT Test Prep in San Diego, GRE Test Prep in Washington DC, MCAT Test Prep in San Diego, LSAT Test Prep in San Francisco-Bay Area, SAT 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 2 : Parametric, Polar, and Vector

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #205 : Vector

Example Question #47 : Vector Calculations

Example Question #206 : Vector

Example Question #207 : Vector

Example Question #208 : Vector

Example Question #51 : Vector Calculations

Example Question #211 : Vector

Example Question #212 : Vector

Example Question #213 : Vector

Example Question #1073 : Calculus Ii

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: