Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Parallel and Perpendicular Vectors in Two Dimensions

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #31 : Determine If Two Vectors Are Parallel Or Perpendicular

Find the angle between the vectors $\small <4, -6>$ and $\small <3, 11>$ .

Possible Answers:

$\small 48.81^o$

$\small 89.54^o$

$\small 83.83^o$

$\small 131.05^o$

$\small 17.97 ^o$

Correct answer:

$\small 131.05^o$

Explanation:

To determine the angle between our two vectors, we can use the fact that for any 2 vectors $\small \vec{a}$ and $\small \vec{b}$ , $\small \vec{a} \cdot \vec{b} = ||\vec{a}||*||\vec{b}||*cos(\theta)$ where $\small ||\vec{a}||$ is the magnitude and $\small \theta$ is the angle between the 2 vectors, which is what we are looking for.

Working from the left, we can first find the dot product, $\small <4, -6> \cdot <3, 11> = 4*3 + -6*11 = 12 -66=-54$

Now we'll find the magnitudes of the two vectors by using the Pythagorean Theorem:

$\small 4^2 + (-6)^2 = C^2$

$\small 16 + 36 = C^2$

$\small 52 = C^2$ take the square root of both sides

$\small \sqrt{52}=C$

$\small 3^2 + 11^2 = C^2$

$\small 9+121 = C^2$

$\small 130 = C^2$

$\small \sqrt{130}=C$

Now we can plug these values back into the equation to start solving for theta:

$\small -54 = \sqrt{52}*\sqrt{130}*cos(\theta)$ multiply the two numbers inside the radicals:

$\small -54 = \sqrt {6,760}*cos(\theta)$ divide both sides by $\small \sqrt{6,760}$

$\small -0.6567 \approx cos(\theta)$ take the inverse cosine of both sides

$\small 131.05 \approx \theta$

Report an Error

Example Question #542 : Pre Calculus

Find the angle between the vectors $\small \small (13, 20^o)$ and $\small <-2, -7>$ . Note that the first vector is in polar form and the second is in component form.

Possible Answers:

$\small \small 125.9 ^ o$

$\small 99.99^o$

$\small 85.4^o$

$\small 5.54 ^ o$

$\small 54.32^o$

Correct answer:

$\small \small 125.9 ^ o$

Explanation:

To find the angle between two vectors, we can use the fact that $\small \vec{a} \cdot \vec{b}= ||\vec{a}||*||\vec{b}||*cos(\theta)$ . In order to find the dot product, we need to convert the vector $\small (13, 20^o)$ to component form. This is easiest to do after drawing a quick sketch of the vector:

Vector 5

To find the vertical component, set up an equation involving sine, since the vertical component is the side of a right triangle across from the 20-degree angle:

$\small sin(20^o)=\frac{y}{13}$ evaluate the sine of 20, then multiply by 13

$\small 4.446 \approx y$

To find the horizontal component, set up an equation involving cosine, since the horizontal component is the side of a right triangle adjacent to the 20-degree angle:

$\small cos(20^o)= \frac{x}{13}$ evaluate the cosine of 20, then multiply by 13

$\small 12.216 \approx x$

The component form of this vector can be written as $\small <12.21, 4.44>$

Next, we can find the magnitude of the vector already in component form by using the Pythagorean Theorem:

$\small (-2)^2 + (-7)^2 = C^2$

$\small 4 + 49 = C^2$

$\small 53 = C^2$ take the square root of both sides

$\small \sqrt{53}=C$

Now we have all the information that we need to solve for theta.

$\small <12.21, 4.44> \cdot <-2, -7> = 13*\sqrt{53}*cos(\theta)$ Find the dot product for the left side:

$\small 12.21*-2 + 4.44*-7 = -55.5$ which we can now put back in the equation:

$\small -55.5 = 13*\sqrt{53}*cos(\theta)$ now divide both sides by $\small 13*\sqrt{53} \approx 94.64$

$\small 0.5864 \approx cos(\theta)$ take invese cosine of both sides

$\small \small 125.90^o \approx \theta$

Report an Error

Example Question #33 : Matrices And Vectors

Find ${} \vec{a} \cdot \vec{b}$ where ${} \vec{a} = <3, 7>$ and $\small \small \vec{b} = (3\sqrt2, 45^o)$ . Note that $\small \vec{a}$ is in component form while $\small \vec{b}$ is in polar form.

Possible Answers:

$\small 315+9\sqrt2$

$\small 9+21\sqrt2$

$\small 30$

$\small 23$

$\small 21+9\sqrt2$

Correct answer:

$\small 30$

Explanation:

First, convert $\small \vec{b}$ to component form. The easiest way to do that would be with special right triangles, knowing that for a 45-45-90 triangle the ratio of the hypotenuse to both of the legs is $\small \sqrt2 : 1$ . So if the magnitude of the vector is $\small 3\sqrt2$ , then the components must both be 3.

This could also be found by setting up and solving the equations:

$\small cos(45)=\frac{x}{3\sqrt2}$ and $\small sin(45)=\frac{y}{3\sqrt2}$

Now that both vectors are component form, we see that we're multiplying

$\small <3,7> \cdot <3,3> = 3*3 + 7*3 = 9+21 = 30$

Report an Error

Example Question #31 : Determine If Two Vectors Are Parallel Or Perpendicular

$\vec{v}=-4i+5j$

$\vec{w}=5i+4j$

Which of the following best explains whether the two vectors above are perpendicular or parallel?

Possible Answers:

Parallel, because their dot product is one.

Perpendicular, because their dot product is one.

Parallel, because their dot product is zero.

Neither perpendicular nor parallel, because their dot product is neither zero nor one.

Perpendicular, because their dot product is zero.

Correct answer:

Perpendicular, because their dot product is zero.

Explanation:

Two vectors are perpendicular if their dot product is zero, and parallel if their dot product is 1.

Take the dot product of our two vectors to find the answer:

$v_i\cdot w_i+v_j\cdot w_j=v \cdot w$

Using our given vectors:

$\vec{v}=-4i+5j$

$\vec{w}=5i+4j$

$v\cdot w=(-4\cdot 5)+(5\cdot 4)=-20+20=0$

Thus our two vectors are perpendicular.

Report an Error

View Pre-Calculus Tutors

Brandon
Certified Tutor

University of Houston, Bachelor of Science, Biology, General. University of North Texas, Doctor of Philosophy, Pharmacy.

View Pre-Calculus Tutors

Chelsea
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Physics. The University of Texas at Dallas, Master of Science, Bioinf...

View Pre-Calculus Tutors

Rachel
Certified Tutor

Grand Canyon University, Bachelor of Science, Mathematics Teacher Education. Grand Canyon University, Master of Science, Addi...

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in Atlanta, MCAT Tutors in New York City, Reading Tutors in Los Angeles, Reading Tutors in Houston, French Tutors in Dallas Fort Worth, ISEE Tutors in Houston, Physics Tutors in Los Angeles, Spanish Tutors in Los Angeles, French Tutors in Philadelphia, GMAT Tutors in Seattle

Popular Courses & Classes

GMAT Courses & Classes in Los Angeles, ACT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Phoenix, Spanish Courses & Classes in Houston, SAT Courses & Classes in Atlanta, ISEE Courses & Classes in Atlanta, MCAT Courses & Classes in New York City, LSAT Courses & Classes in Boston, SSAT Courses & Classes in Miami, SSAT Courses & Classes in New York City

Popular Test Prep

SAT Test Prep in San Diego, ACT Test Prep in Denver, ISEE Test Prep in Miami, LSAT Test Prep in San Diego, SAT Test Prep in Los Angeles, SAT Test Prep in San Francisco-Bay Area, ISEE Test Prep in San Francisco-Bay Area, GRE Test Prep in Atlanta, GRE Test Prep in San Francisco-Bay Area, SSAT Test Prep in Boston

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

Precalculus : Parallel and Perpendicular Vectors in Two Dimensions

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #31 : Determine If Two Vectors Are Parallel Or Perpendicular

Example Question #542 : Pre Calculus

Example Question #33 : Matrices And Vectors

Example Question #31 : Determine If Two Vectors Are Parallel Or Perpendicular

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: