We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

Calculus 3 : Vectors and Vector Operations

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 #161 : Vectors And Vector Operations

Find the cross product of

$\left \langle x, \cos(z), y^2\right \rangle, \left \langle 3, 4, 6\right \rangle$

written in vector form

Possible Answers:

$\left \langle 6\cos(z)-4y^2, 3y^2-6x, 4x-3\cos(z)\right \rangle$

$\left \langle 6\cos(z)+4y^2, 3y^2+6x, 4x+3\cos(z)\right \rangle$

$\left \langle 6\cos(z), 3y^2-6x, 4x-3\cos(z)\right \rangle$

$\left \langle -6\cos(z)+4y^2, -3y^2+6x, -4x+3\cos(z)\right \rangle$

Correct answer:

$\left \langle 6\cos(z)-4y^2, 3y^2-6x, 4x-3\cos(z)\right \rangle$

Explanation:

We first write the determinant in order to take the cross product of the two vectors:

$\begin{vmatrix} \hat{i}& \hat{j}& \hat{k}\\ x&\cos(z) &y^2 \\ 3&4 & 6 \end{vmatrix}$

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:

$6\cos(z)\hat{i}+4x\hat{k}+3y^2\hat{j}-(3\cos(z)\hat{k}+4y^2\hat{i}+6x\hat{j})=(6\cos(z)-4y^2)\hat{i}+(3y^2-6x)\hat{j}+(4x-3\cos(z))\hat{k}$

which written as a vector becomes

$\left \langle 6\cos(z)-4y^2, 3y^2-6x, 4x-3\cos(z)\right \rangle$

Report an Error

Example Question #162 : Vectors And Vector Operations

Find the cross product of the two vectors, written in standard form:

$\left \langle 3y, 0, 5z\right \rangle, \left \langle 4, 1, 2\right \rangle$

Possible Answers:

$-5z\hat{i}+(20z-6y)\hat{j}+3y\hat{k}$

-5z+20z-3y

$-5z\hat{i}+20z\hat{j}-3y\hat{k}$

$5z\hat{i}+(20z+6y)\hat{j}+3y\hat{k}$

Correct answer:

$-5z\hat{i}+(20z-6y)\hat{j}+3y\hat{k}$

Explanation:

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

$\begin{vmatrix} \hat{i}&\hat{j} &\hat{k} \\ 3y&0 &5z \\ 4& 1& 2 \end{vmatrix}$

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

$3y\hat{k}+20z\hat{j}-(5z\hat{i}+6y\hat{j})=-5z\hat{i}+(20z-6y)\hat{j}+3y\hat{k}$

Report an Error

Example Question #163 : Vectors And Vector Operations

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

$\left \langle xy, 3z, z\right \rangle X \left \langle 5, 0, 1\right \rangle$

Possible Answers:

$\left \langle 3z, 5z+xy, 15z\right \rangle$

$\left \langle 3z, 5z+xy, -15z\right \rangle$

$\left \langle 3z, 5z+xyz, -15z\right \rangle$

$\left \langle 3z, 5z-xy, -15z\right \rangle$

Correct answer:

$\left \langle 3z, 5z-xy, -15z\right \rangle$

Explanation:

We first must write the determinant in order to take the cross product of the two vectors:

$\begin{vmatrix} \hat{i}&\hat{j} &\hat{k} \\ xy& 3z& z\\ 5& 0& 1 \end{vmatrix}$

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

$3z\hat{i}+5z\hat{j}-(15z\hat{k}+xy\hat{j})=3z\hat{i}+(5z-xy)\hat{j}-15z\hat{k}=\left \langle 3z, 5z-xy, -15z\right \rangle$

Report an Error

Example Question #164 : Vectors And Vector Operations

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

$\left \langle x^2, y^2, z\right \rangle, \left \langle 0, 1, 4\right \rangle$

Possible Answers:

$(4y^2-z)\hat{i}-4x\hat{j}$

$(4y^2-z)\hat{i}+x^2\hat{k}$

$(4y^2+z)\hat{i}+4x\hat{j}+x^2\hat{k}$

$(4y^2-z)\hat{i}-4x\hat{j}+x^2\hat{k}$

Correct answer:

$(4y^2-z)\hat{i}-4x\hat{j}+x^2\hat{k}$

Explanation:

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

$\begin{vmatrix} \hat{i}&\hat{j} &\hat{k} \\ x^2&y^2 &z \\ 0&1 &4 \end{vmatrix}$

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

$4y^2\hat{i}+x^2\hat{k}-(z\hat{i}+4x^2\hat{j})=(4y-2)\hat{i}-4x^2\hat{j}+x^2\hat{k}$

Report an Error

Example Question #165 : Vectors And Vector Operations

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

$\left \langle 2, 4, 4\right \rangle,\left \langle x, y, z\right \rangle$

Possible Answers:

$\left \langle 4z-4y, 4x-4, 2y-4x\right \rangle$

$\left \langle 4z+4y, 4x+4, 2y+4x\right \rangle$

$\left \langle 2z-2y, 2x-2, y-2x\right \rangle$

4z-2y-4

Correct answer:

$\left \langle 4z-4y, 4x-4, 2y-4x\right \rangle$

Explanation:

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

$\begin{vmatrix} \hat{i}&\hat{j} &\hat{k} \\ 2&4 &4 \\ x& y& z \end{vmatrix}$

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

$4z\hat{i}+2y\hat{k}+4x\hat{j}-(4x\hat{k}+4y\hat{i}+4\hat{j})=(4z-4y)\hat{i}+(4x-4)\hat{j}+(2y-4x)\hat{k}=\left \langle 4z-4y, 4x-4, 2y-4x\right \rangle$

Report an Error

Example Question #166 : Vectors And Vector Operations

Find the cross product between the vectors $\left \langle 3,-1,-2\right \rangle$ and $\left \langle 4,-2,0\right \rangle$

Possible Answers:

$\left \langle 1,5,4\right \rangle$

$\left \langle 4,8,-2\right \rangle$

$\left \langle 4,6,2\right \rangle$

Correct answer:

$\left \langle 4,8,-2\right \rangle$

Explanation:

To find the cross product between vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , you find the determinant of the 3x3 matrix $\begin{bmatrix} i& j&k \\ x_1&y_1 &z_1 \\ x_2&y_2 &z_2 \end{bmatrix}$ . The determinant in this case is $i(0+4)-j(0-8)+k(-6+4)=\left \langle 4,8,-2\right \rangle$

Report an Error

Example Question #167 : Vectors And Vector Operations

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

Possible Answers:

$\left \langle 2,5,-2\right \rangle$

$\left \langle -1,-7,-17\right \rangle$

$\left \langle 2,-4,5\right \rangle$

$\left \langle -1,7,-17\right \rangle$

Correct answer:

$\left \langle -1,7,-17\right \rangle$

Explanation:

To find the cross product between vectors $a=\left \langle x_1,y_1,z_1\right \rangle$ and $b=\left \langle x_2,y_2,z_2\right \rangle$ , you find the determinant of the 3x3 matrix $\begin{bmatrix} i& j&k \\ x_1&y_1 &z_1 \\ x_2&y_2 &z_2 \end{bmatrix}$ . The determinant in this case is $i(-5+4)-j(-1-6)+k(-2-15)=\left \langle -1,7,-17\right \rangle$

Report an Error

Example Question #168 : Vectors And Vector Operations

Two vectors u and v and their cross product have the following magnitudes:

$|\textbf{u}|=2\sqrt{2}\:,\: \: |\textbf{v}|=4, \:\: |\textbf{u}\times \textbf{v}|=8$

What is the angle between the two vectors?

Possible Answers:

$90\degree$

$30\degree$

$45\degree$

$20\degree$

$60\degree$

Correct answer:

$45\degree$

Explanation:

The magnitude of the cross product of two vectors u and v can be defined by the cross product and the angle theta between them as follows:

$|\textbf{u}\times\textbf{v}|=|\textbf{u}||\textbf{v}|\sin\theta$

Solving for theta and substituting the give quantities, we obtain:

$\theta=\sin^{-1}\left ( \frac{|\mathbf{u\times v|}}{|\mathbf{u}||\mathbf{v}|}\right )= \sin^{-1}\left ( \frac{8}{2\sqrt{2}\cdot 4}\right )=\sin^{-1}\left ( \frac{1}{\sqrt{2}}\right ) =45\degree$

Report an Error

Example Question #169 : Vectors And Vector Operations

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

$\left \langle z, 3y, x^2\right \rangle \times \left \langle 9, 0, 0\right \rangle$

Possible Answers:

$\left \langle 0, 9x^2, 27y\right \rangle$

$\left \langle 0, 9x^2, -27y\right \rangle$

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

$\left \langle 9x^2,0, -27y\right \rangle$

Correct answer:

$\left \langle 0, 9x^2, -27y\right \rangle$

Explanation:

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

$\begin{vmatrix} \hat{i}&\hat{j} &\hat{k} \\ z& 3y&x^2 \\ 9&0 &0 \end{vmatrix}$

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

$9x^2\hat{j}-(27y\hat{k})=\left \langle 0, 9x^2, -27y\right \rangle$

Report an Error

Example Question #170 : Vectors And Vector Operations

Find the cross product between the vectors $\left \langle 4,5,1\right \rangle$ and $\left \langle 2,0,1\right \rangle$

Possible Answers:

$\left \langle 5,-2,10\right \rangle$

$\left \langle 5,2,10\right \rangle$

$\left \langle 1,-2,5\right \rangle$

$\left \langle 3,-5,1\right \rangle$

Correct answer:

$\left \langle 5,-2,10\right \rangle$

Explanation:

To find the cross product between the vectors, we find the determinant of the 3x3 matrix $\begin{bmatrix} i& j& k\\ a&b &c \\ x&y &z \end{bmatrix}$ , where one vector is $\left \langle a,b,c\right \rangle$ and the other is $\left \langle x,y,z\right \rangle$ .

Using the formula for the determinant as determinant=i(bz-yc)-j(az-xc)+k(ay-xb) . we get:

$i(5-0)-j(4-2)+k(0-10)=\left \langle 5,-2,-10\right \rangle$

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

Kurosh
Certified Tutor

University of tehran, Bachelor of Engineering, Electrical Engineering.

View Calculus 3 Tutors

Dalton
Certified Tutor

University of Massachusetts Amherst, Doctor of Philosophy, Mathematics.

All Calculus 3 Resources

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

Popular Subjects

Reading Tutors in Los Angeles, GMAT Tutors in Boston, ISEE Tutors in San Diego, Biology Tutors in Chicago, English Tutors in Philadelphia, GRE Tutors in Dallas Fort Worth, MCAT Tutors in Miami, Spanish Tutors in Philadelphia, Physics Tutors in San Francisco-Bay Area, SAT Tutors in Denver

Popular Courses & Classes

SSAT Courses & Classes in Miami, ISEE Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Miami, GRE Courses & Classes in Phoenix, GRE Courses & Classes in Atlanta, SAT Courses & Classes in Atlanta, LSAT Courses & Classes in Miami, ISEE Courses & Classes in Phoenix, Spanish Courses & Classes in Los Angeles

Popular Test Prep

SSAT Test Prep in Boston, ACT Test Prep in Los Angeles, MCAT Test Prep in Houston, LSAT Test Prep in Miami, LSAT Test Prep in Denver, SSAT Test Prep in Washington DC, LSAT Test Prep in Phoenix, ACT Test Prep in Chicago, GRE Test Prep in Seattle, MCAT Test Prep in San Diego

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 : Vectors and Vector Operations

Study concepts, example questions & explanations for Calculus 3

All Calculus 3 Resources

Example Questions

Example Question #161 : Vectors And Vector Operations

Example Question #162 : Vectors And Vector Operations

Example Question #163 : Vectors And Vector Operations

Example Question #164 : Vectors And Vector Operations

Example Question #165 : Vectors And Vector Operations

Example Question #166 : Vectors And Vector Operations

Example Question #167 : Vectors And Vector Operations

Example Question #168 : Vectors And Vector Operations

Example Question #169 : Vectors And Vector Operations

Example Question #170 : Vectors And Vector Operations

All Calculus 3 Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: