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 #2 : Vector Calculations

Evaluate the dot product of $\left \langle 1,6\right \rangle$ and $\left \langle 1,6\right \rangle$ .

Possible Answers:

$\infty$

Correct answer:

Explanation:

Let vectors $\vec{a}=\left \langle a1,a2\right \rangle$ and $\vec{b}=\left \langle b1,b2\right \rangle$ .

The formula for the dot product is:

$\vec{a}\cdot\vec{b} = a1\cdot b1+ a2\cdot b2$

Follow this formula and simplify.

$\left \langle 1,6\right \rangle\cdot\left \langle 1,6\right \rangle = (1)(1)+(6)(6)=37$

Report an Error

Example Question #2 : Vector Calculations

Solve: $\left \langle 3,4,5\right \rangle\cdot\left \langle1,2,3 \right \rangle$

Possible Answers:

$\left \langle 2,-4,2\right \rangle$

$\left \langle 3,8,15\right \rangle$

$\left \langle 4,6,8\right \rangle$

Correct answer:

Explanation:

The problem $\left \langle 3,4,5\right \rangle\cdot\left \langle1,2,3 \right \rangle$ is in the form of a dot product. The final answer must be an integer, and not in vector form.

Write the formula for the dot product.

$\left \langle a_1,a_2,a_3\right \rangle\cdot\left \langle b_1,b_2,b_3\right \rangle=a_1b_1+a_2b_2+a_3b_3$

Substitute the givens and solve.

(3)(1)+(4)(2)+(5)(3)=3+8+15=26

Report an Error

Example Question #8 : Vector Calculations

Suppose $\vec{a}= \left \langle 2,4,-3\right \rangle$ . Find the magnitude of $2\vec{a}$ .

Possible Answers:

$\sqrt{69}$

$2\sqrt{29}$

$\sqrt{29}$

Correct answer:

$2\sqrt{29}$

Explanation:

Calculate $2\vec{a}$ .

$\vec{a}= \left \langle 2,4,-3\right \rangle$

$2\vec{a}= \left \langle 4,8,-6\right \rangle$

Find the magnitude.

$|2\vec{a}|=\sqrt{4^2+8^2+(-6)^2}=\sqrt{16+64+36}=\sqrt{116}=2\sqrt{29}$

Report an Error

Example Question #161 : Vector

Two particles move freely in two dimensional space. The first particle's location as a function of time is $\small \left<3t^{2}-t,4\right>$ , and the second particle's location is $\small \small \left<6,t+1/t\right>$ . Will the particles ever collide for $\small t>0$ ?

Possible Answers:

Impossible to determine

No, because the particles' $\small x$ and $\small y$ coordinates are never the same simultaneously at any instant in time.

Yes, because the particles have the same $\small x$ or $\small y$ component (not necessarily simultaneously).

No, because the particles never have the same $\small x$ or $\small y$ component (not necessarily simultaneously).

Yes, because the particles' $\small x$ and $\small y$ coordinates are the same simultaneously at a certain instant in time.

Correct answer:

No, because the particles' $\small x$ and $\small y$ coordinates are never the same simultaneously at any instant in time.

Explanation:

In order for the particles to collide, their $\small x$ and $\small y$ coordinates must be equal simultaneously. In order to check if this happens, we can set the particles' $\small x$ -coordinates and $\small y$ -coordinates equal to each other.

Let's start with the $\small x$ -coordinate:

$\small \small \small 3t^{2}-t=6\rightarrow 3t^2-t-6=0$ .

Using the quadratic formula

$\small \small t=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ , we get $\small \small t=\frac{1+\sqrt{73}}{6}\approx1.59$

the other root is negative, so it can be discarded since $\small t>0$ .

Now let's do the $\small y$ -coordinate: $\small \small t+1/t=4\rightarrow t^2+1=4t\rightarrow t^2-4t+1=0$ . Use the quadratic formula again to solve for $\small t$ , and you'll get $\small 2\pm\sqrt{3}$ (these roots are approximately $\small 0.27$ and $\small 3.73$ ). The particles never have the same $\small x$ and $\small y$ coordinate simultaneously, so they do not collide.

Report an Error

Example Question #1 : Vector Calculations

$\vec{a}=<3,2,1>$

$\vec{b}=<1,0,10>$

Calculate $\vec{a}\cdot\vec{b}$

Possible Answers:

Correct answer:

Explanation:

$\vec{a}=<3,2,1>$

$\vec{b}=<1,0,10>$

$\vec{a}\cdot\vec{b}$ is simply the dot product of these two vectors. Mathematically, this is calculated as follows.

$(3\cdot 1)+(2\cdot 0)+(1\cdot 10)$ =13

Report an Error

Example Question #11 : Vector Calculations

Find the dot product of $a=\left \langle 3,4,7\right \rangle$ and $b=\left \langle -1,5,6\right \rangle$ .

Possible Answers:

Correct answer:

Explanation:

To find the dot product of $a=\left \langle 3,4,7\right \rangle$ and $b=\left \langle -1,5,6\right \rangle$ , calculate the sum of the products of the vectors' corresponding components:

$\left \langle 3,4,7\right \rangle\times \left \langle -1,5,6\right \rangle$

$=(3\times -1)+ (4\times 5)+(7\times 6)$

=-3+ 20+42

=-3+ 62

=59

Report an Error

Example Question #11 : Vector Calculations

Find the dot product of $a=\left \langle 5,-5,2\right \rangle$ and $b=\left \langle 2,3,8\right \rangle$ .

Possible Answers:

Correct answer:

Explanation:

To find the dot product of $a=\left \langle 5,-5,2\right \rangle$ and $b=\left \langle 2,3,8\right \rangle$ , calculate the sum of the products of the vectors' corresponding components:

$\left \langle 5,-5,2\right \rangle\times \left \langle 2,3,8\right \rangle$

$=(5\times 2)+ (-5\times 3)+(2\times 8)$

=10+ (-15)+16

=10+ 1

=11

Report an Error

Example Question #12 : Vector Calculations

Find the dot product of $a=\left \langle 9,-8,7\right \rangle$ and $b=\left \langle 4,-1,2\right \rangle$ .

Possible Answers:

Correct answer:

Explanation:

To find the dot product of $a=\left \langle 9,-8,7\right \rangle$ and $b=\left \langle 4,-1,2\right \rangle$ , calculate the sum of the products of the vectors' corresponding components:

$\left \langle 9,-8,7\right \rangle\times \left \langle 4,-1,2\right \rangle$

$=(9\times 4)+ (-8\times -1)+ (7\times 2)$

=36+ 8+ 14

=36+ 22

=58

Report an Error

Example Question #511 : Parametric, Polar, And Vector

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

Possible Answers:

Correct answer:

Explanation:

The dot product of two vectors is the sum of the products of their composite elements. Given $a=\left \langle 2,5,-4\right \rangle$ and $b=\left \langle 1,7,5\right \rangle$ , the dot product would therefore be:

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

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

=2+35+(-20)

=37+(-20)

=17

Report an Error

Example Question #512 : Parametric, Polar, And Vector

Find the dot product of $a=\left \langle 6,7,-3\right \rangle$ and $b=\left \langle -4,-5,10\right \rangle$ .

Possible Answers:

Correct answer:

Explanation:

The dot product of two vectors is the sum of the products of their composite elements. Given $a=\left \langle 6,7,-3\right \rangle$ and $b=\left \langle -4,-5,10\right \rangle$ , the dot product would therefore be:

$\left \langle 6,7,-3\right \rangle\times \left \langle -4,-5,10\right \rangle$

$=(6\times -4)+(7\times -5)+(-3\times 10)$

=(-24)+(-35)+(-30)

=(-24)+(-65)

=-89

Report an Error

View Calculus 2 Tutors

Rohan
Certified Tutor

Purdue University-Main Campus, Bachelor of Engineering, Aerospace Engineering.

View Calculus 2 Tutors

David
Certified Tutor

American University of Armenia, Bachelor, Computer Science.

View Calculus 2 Tutors

Samita
Certified Tutor

University of Houston, Bachelor of Science, Computer Science. University of Houston, Non Degree Bachelors, Biology, General.

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, ACT Tutors in Seattle, Computer Science Tutors in Washington DC, Math Tutors in Dallas Fort Worth, Math Tutors in San Francisco-Bay Area, Physics Tutors in Los Angeles, MCAT Tutors in Seattle, Spanish Tutors in Houston, Computer Science Tutors in Seattle, SSAT Tutors in New York City

Popular Courses & Classes

GMAT Courses & Classes in Miami, GRE Courses & Classes in Washington DC, GRE Courses & Classes in San Francisco-Bay Area, ISEE Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in San Diego, SAT Courses & Classes in Philadelphia, ACT Courses & Classes in Chicago, LSAT Courses & Classes in Phoenix, GRE Courses & Classes in New York City, ACT Courses & Classes in Boston

Popular Test Prep

SAT Test Prep in Houston, MCAT Test Prep in Washington DC, GRE Test Prep in Philadelphia, ACT Test Prep in Miami, SSAT Test Prep in Los Angeles, SSAT Test Prep in Phoenix, SSAT Test Prep in Houston, GRE Test Prep in Dallas Fort Worth, GRE Test Prep in Phoenix, SAT Test Prep in New York City

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 #2 : Vector Calculations

Example Question #2 : Vector Calculations

Example Question #8 : Vector Calculations

Example Question #161 : Vector

Example Question #1 : Vector Calculations

Example Question #11 : Vector Calculations

Example Question #11 : Vector Calculations

Example Question #12 : Vector Calculations

Example Question #511 : Parametric, Polar, And Vector

Example Question #512 : 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: