We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 2 : Calculus II

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 #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:

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 $a=\left \langle 5,7,2\right \rangle$ 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

Example Question #1074 : Calculus Ii

What is the cross product of $a=\left \langle 9,8,7\right \rangle$ and $b=\left \langle 0,1,2\right \rangle$ ?

Possible Answers:

$\left \langle -9, 18, 9 \right \rangle$

$\left \langle -9, -18, -9 \right \rangle$

$\left \langle -9, -18, 9 \right \rangle$

$\left \langle -9, 18, -9 \right \rangle$

$\left \langle 9, 18, 9 \right \rangle$

Correct answer:

$\left \langle 9, 18, 9 \right \rangle$

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 $a=\left \langle 9,8,7\right \rangle$ and $b=\left \langle 0,1,2\right \rangle$ , the cross product $a \times b$ is:

$\begin{Bmatrix} i&j &k \\ 9&8 &7 \\ 0&1 &2 \end{Bmatrix}$

$=\left \langle (8)(2)-(7)(1), (9)(2)-(7)(0), (9)(1)-(8)(0) \right \rangle$

$=\left \langle 16-7, 18-0, 9-0 \right \rangle$

$=\left \langle 9, 18, 9 \right \rangle$

Report an Error

Example Question #1075 : Calculus Ii

What is the cross product of $a=\left \langle 7,-4,3\right \rangle$ and $b=\left \langle 5,-5,5\right \rangle$ ?

Possible Answers:

$\left \langle 5, 20, -15 \right \rangle$

$\left \langle -5,-20, -15 \right \rangle$

$\left \langle -5, 20, 15 \right \rangle$

$\left \langle -5, 20, -15 \right \rangle$

$\left \langle 5, 20, 15 \right \rangle$

Correct answer:

$\left \langle -5, 20, -15 \right \rangle$

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 $a=\left \langle 7,-4,3\right \rangle$ and $b=\left \langle 5,-5,5\right \rangle$ , the cross product $a \times b$ is:

$\begin{Bmatrix} i&j &k \\ 7&-4 &3 \\ 5&-5 &5 \end{Bmatrix}$

$=\left \langle (-4)(5)-(3)(-5), (7)(5)-(3)(5), (7)(-5)-(-4)(5) \right \rangle$

$=\left \langle -20+15, 35-15, -35+20 \right \rangle$

$=\left \langle -5, 20, -15 \right \rangle$

Report an Error

Example Question #1076 : Calculus Ii

What is the cross product of $a=\left \langle 0,-3,3\right \rangle$ and $b=\left \langle -3,0,3\right \rangle$ ?

Possible Answers:

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

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

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

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

$\left \langle 9, 9,9 \right \rangle$

Correct answer:

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

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 $a=\left \langle 0,-3,3\right \rangle$ and $b=\left \langle -3,0,3\right \rangle$ , the cross product $a \times b$ is:

$\begin{Bmatrix} i&j &k \\ 0&-3 &3 \\ -3&0 &3 \end{Bmatrix}$

$=\left \langle (-3)(3)-(3)(0), (0)(3)-(3)(-3), (0)(0)-(-3)(-3) \right \rangle$

$=\left \langle -9-0, 0+9, 0-9 \right \rangle$

$=\left \langle -9, 9,-9 \right \rangle$

Report an Error

Example Question #214 : Vector

What is the dot product of $a=\left \langle 3,5,7\right \rangle$ and $b=\left \langle -4,-1,2\right \rangle$ ?

Possible Answers:

Correct answer:

Explanation:

The dot product of two vectors is the sum of the products of the vectors' corresponding elements. Given $a=\left \langle 3,5,7\right \rangle$ and $b=\left \langle -4,-1,2\right \rangle$ , then:

$\left \langle 3,5,7\right \rangle\times \left \langle -4,-1,2\right \rangle$

$=(3\times -4)+(5\times -1)+(7\times 2)$

=(-12)+(-5)+(14)

=-17+14

=-3

Report an Error

Example Question #215 : Vector

What is the dot product of $a=\left \langle 2,0,-2\right \rangle$ and $b=\left \langle 6,1,7\right \rangle$ ?

Possible Answers:

Correct answer:

Explanation:

The dot product of two vectors is the sum of the products of the vectors' corresponding elements. Given $a=\left \langle 2,0,-2\right \rangle$ and $b=\left \langle 6,1,7\right \rangle$ then:

$\left \langle 2,0,-2\right \rangle\times \left \langle 6,1,7\right \rangle$

$=(2\times 6)+(0\times 1)+(-2\times7)$

=12+0-14

=-2

Report an Error

Example Question #216 : Vector

What is the dot product of $a=\left \langle 0,5,1\right \rangle$ and $b=\left \langle 5,1,0\right \rangle$ ?

Possible Answers:

Correct answer:

Explanation:

The dot product of two vectors is the sum of the products of the vectors' corresponding elements. Given $a=\left \langle 0,5,1\right \rangle$ and $b=\left \langle 5,1,0\right \rangle$ , then:

$\left \langle 0,5,1\right \rangle\times\left \langle 5,1,0\right \rangle$

$=(0\times5)+(5\times1)+(1\times0)$

=0+5+0

Report an Error

Example Question #561 : Parametric, Polar, And Vector

What is the norm of $a=\left \langle 7,2,1\right \rangle$ ?

Possible Answers:

$3\sqrt{10}$

$3\sqrt{6}$

$3\sqrt{7}$

$3\sqrt{3}$

$3\sqrt{5}$

Correct answer:

$3\sqrt{6}$

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 $a=\left \langle 7,2,1\right \rangle$ , then:

$\left \| a\right \|=\sqrt{(7)^{2}+(2)^{2}+(1)^{2}}$

$\left \| a\right \|=\sqrt{49+4+1}$

$\left \| a\right \|=\sqrt{49+5}$

$\left \| a\right \|=\sqrt{54}$

$\left \| a\right \|=\sqrt{9\times6}$

$\left \| a\right \|=3\sqrt{6}$

Report an Error

View Calculus 2 Tutors

Ping
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Computer Science. University of North Texas, Master of Arts, Education.

View Calculus 2 Tutors

MrG
Certified Tutor

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

View Calculus 2 Tutors

Oliver
Certified Tutor

Bard College, Bachelor of Science, Mathematics.

All Calculus 2 Resources

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

Popular Subjects

Reading Tutors in San Diego, Computer Science Tutors in Los Angeles, MCAT Tutors in Philadelphia, Reading Tutors in Dallas Fort Worth, Statistics Tutors in Dallas Fort Worth, Spanish Tutors in Philadelphia, SAT Tutors in Boston, ACT Tutors in New York City, Calculus Tutors in Miami, LSAT Tutors in Phoenix

Popular Courses & Classes

MCAT Courses & Classes in New York City, SSAT Courses & Classes in Chicago, GRE Courses & Classes in Phoenix, SSAT Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in Denver, ACT Courses & Classes in Philadelphia, LSAT Courses & Classes in Denver, SSAT Courses & Classes in Seattle, LSAT Courses & Classes in Atlanta, ISEE Courses & Classes in Houston

Popular Test Prep

ISEE Test Prep in Denver, MCAT Test Prep in Miami, LSAT Test Prep in Atlanta, ISEE Test Prep in Washington DC, ACT Test Prep in New York City, GRE Test Prep in Chicago, LSAT Test Prep in Denver, MCAT Test Prep in Chicago, MCAT Test Prep in Washington DC, ISEE Test Prep in Dallas Fort Worth

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 : Calculus II

Study concepts, example questions & explanations for Calculus 2

All Calculus 2 Resources

Example Questions

Example Question #212 : Vector

Example Question #213 : Vector

Example Question #1073 : Calculus Ii

Example Question #1074 : Calculus Ii

Example Question #1075 : Calculus Ii

Example Question #1076 : Calculus Ii

Example Question #214 : Vector

Example Question #215 : Vector

Example Question #216 : Vector

Example Question #561 : Parametric, Polar, And Vector

All Calculus 2 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: