Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Matrices and Vectors

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 #1 : Solve A System Of Equations In Three Variables Using Augmented Matrices

$A = \begin{bmatrix} 0.7 & -0.4 \\ -0.8 & 0.2 \end{bmatrix}$ and $B= \begin{bmatrix} 0. 3 & 0.4 \\ 0.8 & 0.8 \end{bmatrix}$ .

True or false: A+ B = I , where is the two-by-two identity matrix.

Possible Answers:

True

False

Correct answer:

True

Explanation:

First, it must be established that A+ B is defined. This is the case if and only and have the same number of rows, which is true, and they have the same number of columns, which is also true. A +B is therefore defined.

Addition of two matrices is performed by adding corresponding elements, so

$A +B= \begin{bmatrix} 0.7 & -0.4 \\ -0.8 & 0.2 \end{bmatrix}+ \begin{bmatrix} 0. 3 & 0.4 \\ 0.8 & 0.8 \end{bmatrix}$

$= \begin{bmatrix} 0.7+0.3 & -0.4+0.4 \\ -0.8+0.8 & 0.2+0.8 \end{bmatrix}$

$= \begin{bmatrix} 1 & 0\\ 0 & 1 \end{bmatrix}$

This is the two-by-two identity . Therefore, A+ B = I .

Report an Error

Example Question #592 : Pre Calculus

Using the Reduced Row Echelon Method, the solution matrix to the system of equations reads

Screen shot 2020 07 30 at 11.31.42 am

Which of the following is the accurate depiction of this matrix?

Possible Answers:

x=8,y=-5,z=-2

x=1,y=0,z=0

x=-8,y=5,z=2

x=-2,y=-5,z=8

x=1,y=1,z=1

Correct answer:

x=8,y=-5,z=-2

Explanation:

Traditionally in Row Reduced Echelon Form (first row's values are 1,0,0; second row's values are 0,1,0; third row's values are 0,0,1), x is depicted in the first row, in the second row, and in the third. Therefore the correct answer is x=8,y=-5,z=-2 .

Report an Error

Example Question #1 : Solve A System Of Equations In Three Variables Using Augmented Matrices

Determine which 3x3 matrix represents the following system of equations:

x+y+z=2;

2x-y+3z=-7;

x+y-z=6.

Possible Answers:

$\left[\begin{array}{ccc|c} 1 & 0 & 0 & 2\\ 0 & 1 & 0 & -7\\ 0 & 0 & 1 & 6 \end{array}\right]$

$\left[\begin{array}{ccc|c} 1 & 1 & 1 & -2\\ 2 & -1 & 3 & 7\\ 1 & 1 & -1 & -6 \end{array}\right]$

$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 2\\ 2 & -1 & 3 & -7\\ 1 & 1 & -1 & 6 \end{array}\right]$

$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 1\\ 2 & -1 & 3 & 1\\ 1 & 1 & -1 & 1 \end{array}\right]$

$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 0\\ 2 & -1 & 3 & 0\\ 1 & 1 & -1 & 0 \end{array}\right]$

Correct answer:

$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 2\\ 2 & -1 & 3 & -7\\ 1 & 1 & -1 & 6 \end{array}\right]$

Explanation:

The system of equations in matrix form is written:

$\left[\begin{array}{ccc|c} 1 & 1 & 1 & 2\\ 2 & -1 & 3 & -7\\ 1 & 1 & -1 & 6 \end{array}\right]$

Beginning with the top equation, insert each coefficient into the top row of the matrix, from left to right. Then, place the 2 on the right hand side of the vertical line (anytime you see this vertical line in a matrix, it is called an augmented matrix. Because the first equation's coefficients are 1, 1, and 1, these form the top row of the matrix, with 2 on the right side of the bar. Then, move on to the second row, which has coefficients 2, -1, and 3. Place these into the second row of the matrix, with -7 on the right hand side of the vertical bar. Finally, take the coefficients from the 3rd equation, 1, 1, and -1, and place these in the bottom row with 6 on the right hand side of the vertical bar.

Report an Error

Example Question #1 : Solve A System Of Equations In Three Variables Using Augmented Matrices

Determine which 3x3 matrix represents the following system of equations:

;

Possible Answers:

$\left [ \begin{bmatrix} 1 & 1& 1& \left | 2 \right |\\ 2& -1 & 3&\left | 7 \right | \\ 1& 1& 1 & \left | 6 \right | \end{bmatrix} \right ]$

$\left [ \begin{bmatrix} 1 & 1& 1& \left | 2 \right |\\ 2& -1 & 3&\left | -7 \right | \\ 1& 1& -1 & \left | 6 \right | \end{bmatrix} \right ]$

$\left [ \begin{bmatrix} 1 & 1& 1& \left | 2 \right |\\ 2& -1 & -3&\left | 7 \right | \\ 1& 1& -1 & \left | 6 \right | \end{bmatrix} \right ]$

$\left [ \begin{bmatrix} 2 & 1& 1& \left | 1 \right |\\ 2& -1 & 3&\left | -7 \right | \\ 6& -1& 1 & \left | 1 \right | \end{bmatrix} \right ]$

Correct answer:

$\left [ \begin{bmatrix} 1 & 1& 1& \left | 2 \right |\\ 2& -1 & 3&\left | -7 \right | \\ 1& 1& -1 & \left | 6 \right | \end{bmatrix} \right ]$

Explanation:

The system of equations in matrix form is written: $\left [ \begin{bmatrix} 1 & 1& 1& \left | 2 \right |\\ 2& -1 & 3&\left | -7 \right | \\ 1& 1& -1 & \left | 6 \right | \end{bmatrix} \right ]$

Report an Error

Example Question #1 : Algebraic Vectors And Parametric Equations

Find the unit vector that is in the same direction as the vector $\vec{v}= [3,6,1]$

Possible Answers:

$\vec{u}_v= \left [ 9,36,1 \right ]$

$\vec{u}_v= \left [ 9,3,1 \right ]$

$\vec{u}_v= \left [ {\frac{3}{\sqrt{46}}},\frac{6}{\sqrt{46}} , \frac{1}{\sqrt{46}} \right ]$

$\vec{u}_v= \left [ {\frac{9}{\sqrt{3}}},\frac{36}{\sqrt{6}} ,1 \right ]$

Correct answer:

$\vec{u}_v= \left [ {\frac{3}{\sqrt{46}}},\frac{6}{\sqrt{46}} , \frac{1}{\sqrt{46}} \right ]$

Explanation:

To find the unit vector in the same direction as a vector, we divide it by its magnitude.

The magnitude of $\vec{v}$ is $\left | \vec{v} \right |= \sqrt{3^2+6^2+1^2}=\sqrt{46}$ .

We divide vector $\vec{v}$ by its magnitude to get the unit vector $\vec{u}_v$ :

$\vec{u}_v= \frac{\vec{v}}{\left | \vec{v} \right |}=\frac{1}{\sqrt{46}}\cdot[3,6,1]$

$\vec{u}_v= \left [ {\frac{3}{\sqrt{46}}},\frac{6}{\sqrt{46}} , \frac{1}{\sqrt{46}} \right ]$

All unit vectors have a magnitude of , so to verify we are correct:

$\left | \vec{u}_v \right | = \sqrt{ (\frac{3}{\sqrt{46}})^2 + (\frac{6}{\sqrt{46}})^2 + (\frac{9}{\sqrt{46}})^2 } = \sqrt{\frac{9}{46} + \frac{36}{46} + \frac{1}{46} } = 1$

Report an Error

Example Question #1 : Find The Unit Vector In The Same Direction As A Given Vector

A unit vector has length .

Given the vector

v=<3,6,44,-5,12>

find the unit vector in the same direction.

Possible Answers:

$<1,2,\frac{44}{3},\frac{-5}{3},4>$

$<\frac{1}{2},1,\frac{22}{3},\frac{-5}{6},2>$

<0.065,0.129,0.949,-0.108,0.259>

<3,6,44,-5,12>

$<\frac{3}{44},\frac{3}{22},1,\frac{-5}{44},\frac{3}{11}>$

Correct answer:

<0.065,0.129,0.949,-0.108,0.259>

Explanation:

First, you must find the length of the vector. This is given by the equation:

$\sqrt{3^2+6^2+44^2+(-5)^2+12^2}=\left \| v\right \|$

Then, dividing the vector by its length gives the unit vector in the same direction.

$\frac{v}{\left \| v\right \|}=\frac{<3,6,44,-5,12>}{\sqrt{2150}}\\ \\ \approx<0.065,0.129,0.949,-0.108,0.259>$

Report an Error

Example Question #2 : Find The Unit Vector In The Same Direction As A Given Vector

Put the vector $\small (1,3)$ in unit vector form.

Possible Answers:

$\small (3,1)$

$\left (\frac{1}{10},\frac{3}{10}\right)$

$\left (\frac{3}{10},\frac{1}{10}\right)$

$\left (\frac{3}{\sqrt{10}},\frac{1}{\sqrt{10}}\right)$

$\left (\frac{1}{\sqrt{10}},\frac{3}{\sqrt{10}}\right)$

Correct answer:

$\left (\frac{1}{\sqrt{10}},\frac{3}{\sqrt{10}}\right)$

Explanation:

To get the unit vector that is in the same direction as the original vector $\small v$ , we divide the vector by the magnitude of the vector.

For $\small (1,3)$ , the magnitude is:

$\sqrt{x^2+y^2}$

$\small \sqrt{1^2+3^2}=\sqrt{10}$ .

This means the unit vector in the same direction of $\small (1,3)$ is,

$\small \frac{1}{\sqrt{10}}(1,3)=\left(\frac{1}{\sqrt{10}},\frac{3}{\sqrt{10}}\right)$ .

Report an Error

Example Question #1 : Find The Unit Vector In The Same Direction As A Given Vector

Find the unit vector of

$\left<3,\,4\right>$ .

Possible Answers:

$\left<1,3\right>$

$\left<1,1\right>$

$\left<1,0\right>$

$\left<\frac{3}{5},\, \frac{4}{5}\right>$

Correct answer:

$\left<\frac{3}{5},\, \frac{4}{5}\right>$

Explanation:

In order to find the unit vector u of a given vector v, we follow the formula

$u=\frac{v}{\left | v\right |}$

Let

$v=\left<a,b\right>$

The magnitude of v follows the formula

$\left | v\right |=\sqrt{a^{2}+b^{2}}$ .

For this vector in the problem

$\left | v\right |=\sqrt{3^{2}+4^{2}}$

$=\sqrt{9+16}$

$=\sqrt{25}$

$\left | v\right |=5$ .

Following the unit vector formula and substituting for the vector and magnitude

$u=\frac{\left< 3,4\right>}{5}$ .

As such,

$u\,=\,\left<\frac{3}{5},\, \frac{4}{5}\right>$ .

Report an Error

Example Question #1 : Find The Unit Vector In The Same Direction As A Given Vector

Find the unit vector of

$\left<5,\,12\right>$

Possible Answers:

$\left<1, \frac{12}{5}\right>$

$\left<\frac{1}{2},\frac{1}{2}\right>$

$\left<2,1\right>$

$\left<\frac{5}{13},\,\frac{12}{13}\right>$

Correct answer:

$\left<\frac{5}{13},\,\frac{12}{13}\right>$

Explanation:

In order to find the unit vector u of a given vector v, we follow the formula

$u=\frac{v}{\left | v\right |}$

Let

$v=\left<a,b\right>$

The magnitude of v follows the formula

$\left | v\right |=\sqrt{a^{2}+b^{2}}$

For this vector in the problem

$\left | v\right |=\sqrt{5^{2}+12^{2}}$

$=\sqrt{25+144}$

$=\sqrt{169}$

$\left | v\right |=13$

Following the unit vector formula and substituting for the vector and magnitude

$u=\frac{\left< 5,12\right>}{13}$

As such,

$u\,=\,\left<\frac{5}{13},\, \frac{12}{13}\right>$

Report an Error

Example Question #1 : Find A Vector Equation When Given Two Points

What is the vector that connects the point (2,9) to (7, 18) ?

Possible Answers:

$8.7, \measuredangle 57.2^{\circ}$

None of the other answers

$12.1, \measuredangle 64.3^{\circ}$

$11.4, \measuredangle 58.4^{\circ}$

$10.3, \measuredangle 60.9^{\circ}$

Correct answer:

$10.3, \measuredangle 60.9^{\circ}$

Explanation:

The first step when solving for a vetor is to find the length of the vector. It can be helpful to visualize the system as a right triangle as seen below:

At this point, we can essentially solve the problem as if we're finding the hypotenuse and angle of a right triangle.

Using the Pythagorean Theorem to find the length of the vector we get:

$v = \sqrt{5^2+9^2} = \sqrt{106} = 10.3$

Now we need to find the angle of the vector. We have all three sides of the triangle, so use the trig function that you are most comfortable with. The tangent function is used below because it uses sides that were given in the problem statement.

$tan(\theta) = \frac{9}{5} = 1.8$

$\theta = tan^{-1}(1.8) = 60.9^{\circ}$

Report an Error

View Pre-Calculus Tutors

Jesus Ricardo
Certified Tutor

Benemerita Universidad Autónoma de Puebla, Bachelor, Physics.

View Pre-Calculus Tutors

Jennifer
Certified Tutor

DeVry Universitys Keller Graduate School of Management-Indiana, Bachelor of Science, Mathematics. Arizona State University, M...

View Pre-Calculus Tutors

Mohammad
Certified Tutor

Rutgers University-New Brunswick, Bachelor of Science, Mathematics. Rutgers University-New Brunswick, Master of Science, Stat...

All Precalculus Resources

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

Popular Subjects

Statistics Tutors in New York City, Chemistry Tutors in Seattle, Math Tutors in Philadelphia, Computer Science Tutors in Philadelphia, Biology Tutors in Houston, Chemistry Tutors in San Francisco-Bay Area, GRE Tutors in Los Angeles, Algebra Tutors in Seattle, Biology Tutors in Miami, French Tutors in New York City

Popular Courses & Classes

ACT Courses & Classes in Denver, SSAT Courses & Classes in Phoenix, ACT Courses & Classes in Boston, GMAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Phoenix, ISEE Courses & Classes in Seattle, GRE Courses & Classes in Seattle, SSAT Courses & Classes in Houston, ACT Courses & Classes in Houston, ACT Courses & Classes in Miami

Popular Test Prep

GMAT Test Prep in Phoenix, ACT Test Prep in Denver, ACT Test Prep in Philadelphia, ACT Test Prep in Miami, GRE Test Prep in Dallas Fort Worth, ACT Test Prep in San Francisco-Bay Area, GRE Test Prep in Philadelphia, ISEE Test Prep in Miami, MCAT Test Prep in Phoenix, MCAT 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

Precalculus : Matrices and Vectors

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Solve A System Of Equations In Three Variables Using Augmented Matrices

Example Question #592 : Pre Calculus

Example Question #1 : Solve A System Of Equations In Three Variables Using Augmented Matrices

Example Question #1 : Solve A System Of Equations In Three Variables Using Augmented Matrices

Example Question #1 : Algebraic Vectors And Parametric Equations

Example Question #1 : Find The Unit Vector In The Same Direction As A Given Vector

Example Question #2 : Find The Unit Vector In The Same Direction As A Given Vector

Example Question #1 : Find The Unit Vector In The Same Direction As A Given Vector

Example Question #1 : Find The Unit Vector In The Same Direction As A Given Vector

Example Question #1 : Find A Vector Equation When Given Two Points

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: