Call Now to Set Up Tutoring:

(888) 888-0446

Linear Algebra : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

Create An Account

All Linear Algebra Resources

4 Diagnostic Tests 108 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 … 81 82 83 84 85 86 87 88 89 90 91 92 Next →

Example Question #27 : Vector Vector Product

$\textbf{u} = (\ln 2 , 1, -1 )$

$\textbf{v} = (2, 1, - 2)$

If $\textbf{u} \cdot \textbf{v} = \ln a$ , then evaluate .

Possible Answers:

a = 64

a = 4e

$a = \frac{4}{e}$

$a = 4e^{3}$

a = 12

Correct answer:

$a = 4e^{3}$

Explanation:

The dot product $\textbf{u} \cdot \textbf{v}$ is equal to the sum of the products of the numbers in corresponding positions, so

$\textbf{u} \cdot \textbf{v} = (\ln 2) \cdot 2 + 1 \cdot 1 + (-1 )(-2)$

$\textbf{u} \cdot \textbf{v} = 2\ln 2+ 1 + 2$

$\textbf{u} \cdot \textbf{v} = 3+ 2\ln 2$

Applying the properties of logarithms:

$\textbf{u} \cdot \textbf{v} = 3+ \ln 2^{2}$

$\textbf{u} \cdot \textbf{v} = 3+ \ln 4$

$\textbf{u} \cdot \textbf{v} = \ln e^{3}+ \ln 4$

$\textbf{u} \cdot \textbf{v} = \ln 4 e^{3}$

Therefore, $a = 4e^{3}$ .

Report an Error

Example Question #28 : Vector Vector Product

$\textbf{u} = (1, x, x^{2}, x ^{3}, x^{4}, x^{5})$

$\textbf{v} = (y^{5}, y^{3} , y, y^{4}, 1, y^{2})$

The expression $\textbf{u} \cdot \textbf{v}$ yields a polynomial of what degree?

Possible Answers:

None of the other choices gives a correct response.

Correct answer:

Explanation:

The dot product $\textbf{u} \cdot \textbf{v}$ is the sum of the products of entries in corresponding positions, so

$\textbf{u} \cdot \textbf{v} = 1 (y^{5}) + x (y^{3})+ x^{2}(y )+ x^{3}(y^{4})+ x^{4} (1)+ x^{5 }(y^{2})$

$\textbf{u} \cdot \textbf{v} = y^{5} + x y^{3}+ x^{2}y + x^{3}y^{4}+ x^{4} + x^{5 }y^{2}$

The degree of a term of a polynomial is the sum of the exponents of its variables; the individual terms have degrees 5, 4, 3, 7, 4, and 7, in that order. the degree of the polynomial is the highest of these, which is 7.

Report an Error

Example Question #29 : Vector Vector Product

A triangle has two sides of length and ; their included angle has measure $\theta$ . The measure of the third side can be obtained from the expression

$\sqrt{\textbf{u} \cdot \textbf{v}}$ ,

where $\textbf{u}= (a , b, 2ab)$ and:

Possible Answers:

$\textbf{v} = (a, b, \cos \theta)$

$\textbf{v} = (b, a, -\cos \theta)$

$\textbf{v} = (a, b, -\sin \theta)$

$\textbf{v} = (b, a, -\sin \theta)$

$\textbf{v} = (a, b, -\cos \theta)$

Correct answer:

$\textbf{v} = (a, b, -\cos \theta)$

Explanation:

Given the lengths and of two sides of a triangle, and the measure of their included angle, $\theta$ , the length of the third side of a triangle can be calculated using the Law of Cosines, which states that

$c^{2} = a^{2}+ b^{2} - 2ab\cos \theta$ .

The dot product $\textbf{u} \cdot \textbf{v}$ is equal to the sum of the products of their corresponding entries, and since $c= \sqrt{\textbf{u} \cdot \textbf{v}}$ , we can substitute $\textbf{u} \cdot \textbf{v}$ for $c^{2}$ :

$\textbf{u} \cdot \textbf{v} = a^{2}+ b^{2} - 2ab\cos \theta$

$\textbf{u} \cdot \textbf{v} = a \cdot a + b \cdot b + 2ab \cdot ( - \cos \theta)$

$\textbf{u} = (a , b , 2ab )$ ; it follows that $\textbf{v} = (a, b, -\cos \theta)$ .

Report an Error

Example Question #30 : Vector Vector Product

and are differentiable functions.

$A = \begin{bmatrix} f &g\end{bmatrix}$

$AB = \begin{bmatrix} \frac{\mathrm{d} }{\mathrm{d} x} \left ( \frac{f}{g} \right )\end{bmatrix}$

Which value of makes this statement true?

Possible Answers:

$B = \begin{bmatrix} - \frac{1}{g^{2}} \cdot \frac{\mathrm{d} g}{\mathrm{d} x} \\ \\ \frac{1}{g^{2}} \cdot \frac{\mathrm{d} f}{\mathrm{d} x} \end{bmatrix}$

$B = \begin{bmatrix} \frac{1}{g^{2}} \cdot \frac{\mathrm{d} g}{\mathrm{d} x} \\ \\- \frac{1}{g^{2}} \cdot \frac{\mathrm{d} f}{\mathrm{d} x} \end{bmatrix}$

$B = \begin{bmatrix} \frac{1}{g^{2}} \cdot \frac{\mathrm{d} f}{\mathrm{d} x} \\ \\ \frac{1}{g^{2}} \cdot \frac{\mathrm{d} g}{\mathrm{d} x} \end{bmatrix}$

$B = \begin{bmatrix} \frac{1}{g^{2}} \cdot \frac{\mathrm{d} g}{\mathrm{d} x} \\ \\ \frac{1}{g^{2}} \cdot \frac{\mathrm{d} f}{\mathrm{d} x} \end{bmatrix}$

$B = \begin{bmatrix} - \frac{1}{g^{2}} \cdot \frac{\mathrm{d} f}{\mathrm{d} x} \\ \\ \frac{1}{g^{2}} \cdot \frac{\mathrm{d} g}{\mathrm{d} x} \end{bmatrix}$

Correct answer:

$B = \begin{bmatrix} - \frac{1}{g^{2}} \cdot \frac{\mathrm{d} g}{\mathrm{d} x} \\ \\ \frac{1}{g^{2}} \cdot \frac{\mathrm{d} f}{\mathrm{d} x} \end{bmatrix}$

Explanation:

Recall the quotient rule of differentiation:

$\frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{f}{g} \right ) =\frac{ \frac{\mathrm{d}f }{\mathrm{d} x} \cdot g-f \cdot \frac{\mathrm{d}g }{\mathrm{d} x} }{g^{2}}$

This can be rewritten as

$\frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{f}{g} \right ) =\frac{-f \cdot \frac{\mathrm{d}g }{\mathrm{d} x}+ \frac{\mathrm{d}f }{\mathrm{d} x} \cdot g }{g^{2}}$

$\frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{f}{g} \right ) =\frac{-f \cdot \frac{\mathrm{d}g }{\mathrm{d} x} }{g^{2}}+\frac{ \frac{\mathrm{d}f }{\mathrm{d} x} \cdot g }{g^{2}}$

$\frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{f}{g} \right ) =f \left (- \frac{ \frac{\mathrm{d}g }{\mathrm{d} x} }{g^{2}} \right )+g\left ( \frac{ \frac{\mathrm{d}f }{\mathrm{d} x} }{g^{2}} \right )$

$\frac{\mathrm{d} }{\mathrm{d} x}\left ( \frac{f}{g} \right ) =f \left (- \frac{1 }{g^{2}} \right ) \frac{\mathrm{d}g }{\mathrm{d} x} +g\left ( \frac{1}{g^{2}} \right ) \frac{\mathrm{d}f }{\mathrm{d} x}$

If $A = \begin{bmatrix} f &g\end{bmatrix}$ and $B = \begin{bmatrix} a \\ b \end{bmatrix}$ ,

then multiply corresponding elements and add the products to get the sole element in :

$AB = \begin{bmatrix} f \cdot a + g \cdot b \end{bmatrix}$

Since we want

$AB = \begin{bmatrix} \frac{\mathrm{d} }{\mathrm{d} x} \left ( \frac{f}{g} \right )\end{bmatrix}=\begin{bmatrix} f \left (- \frac{1 }{g^{2}} \right ) \frac{\mathrm{d}g }{\mathrm{d} x} +g\left ( \frac{1}{g^{2}} \right ) \frac{\mathrm{d}f }{\mathrm{d} x} \end{bmatrix}$ ,

It follows that, of the given choices, $a= \left (- \frac{1 }{g^{2}} \right ) \frac{\mathrm{d}g }{\mathrm{d} x}$ and $b= \left ( \frac{1}{g^{2}} \right ) \frac{\mathrm{d}f }{\mathrm{d} x}$ , and

$B = \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} - \frac{1}{g^{2}} \cdot \frac{\mathrm{d} g}{\mathrm{d} x} \\ \\ \frac{1}{g^{2}} \cdot \frac{\mathrm{d} f}{\mathrm{d} x} \end{bmatrix}$ .

Report an Error

Example Question #841 : Linear Algebra

$\textbf{u} = (- 3, 4, 1, 0)$

$\textbf{v} = (0, 1, -2, 4)$

Calculate the angle (nearest degree) between $\textbf{u}$ and $\textbf{v}$ .

Possible Answers:

The angle is undefined, since the vectors are in $\mathbb{R}^{4}$ .

$105^{\circ }$

$175^{\circ }$

$5^{\circ }$

$85^{\circ }$

Correct answer:

$85^{\circ }$

Explanation:

$\textbf{u} = (- 3, 4, 1, 0)$

$\textbf{v} = (0, 1, -2, 4)$

The angle $\; \theta$ between vectors $\textbf{u}$ and $\textbf{v}$ can be calculated using the formula

$\cos \theta = \frac{\textbf{u} \cdot \textbf{v}}{\left \| \textbf{u} \right \|\left \| \textbf{v} \right \|}$ .

$\textbf{u} \cdot \textbf{v}$ , the dot product, is the sum of the products of corresponding entries:

$\textbf{u} \cdot \textbf{v} = -3(0)+4(1)+1(-2)+0(4) = 0 + 4 + (-2)+ 0 = 2$

$\left \| \textbf{u} \right \|$ , the norm of $\textbf{u}$ , is the square root of the sum of the squares of its entries; $\left \| \textbf{v} \right \|$ is defined similarly:

$\left \| \textbf{u} \right \|= \sqrt{ (- 3)^{2}+ 4^{2} + 1^{2} +0^{2}} = \sqrt{9+16+1+0} = \sqrt{26}$

$\left \| \textbf{v} \right \|= \sqrt{ 0^{2}+ 1^{2} +(-2)^{2} +4^{2}} = \sqrt{0+1+4+16} = \sqrt{21}$

$\cos \theta = \frac{2}{ \sqrt{26} \cdot \sqrt{21}} \approx 0.0856$

$\theta \approx \cos ^{-1}0.0856 \approx 85^{\circ }$

Report an Error

Example Question #32 : Vector Vector Product

, , and give the length, width, and height of a rectangular prism.

$\textbf{u} = (L, W, H )$ and $\textbf{v} = (W, H, L )$ .

True or false: $\textbf{u} \cdot \textbf{v}$ gives the surface area of the prism.

Possible Answers:

True

False

Correct answer:

False

Explanation:

The dot product $\textbf{u} \cdot \textbf{v}$ can be calculated by adding the products of the elements in corresponding locations, so

$\textbf{u} \cdot \textbf{v} = LW + WH + LH$ .

The surface area of the prism, , can be found by using the formula:

$A = 2 (LW + WH + LH) = 2( \textbf{u} \cdot \textbf{v})$

Equivalently, $\textbf{u} \cdot \textbf{v}$ gives half the surface area of the prism. The statement is false.

Report an Error

Example Question #33 : Vector Vector Product

$\textbf{u},\textbf{v} , \textbf{w}, \textbf{x} \in \mathbb{R}^{3}$

Which of the following applies to $(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x})$ , where " $\cdot$ " and " $\times$ " refer to the dot product and the cross product of two vectors?

Possible Answers:

$(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x}) \in \mathbb{R}$

$(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x}) \in \mathbb{R}^{3}$

$(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x})$ is an undefined expression.

$(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x}) \in M_{2 \times 2}$

Correct answer:

$(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x}) \in \mathbb{R}^{3}$

Explanation:

The cross product of two vectors in $\mathbb{R}^{3}$ is also a vector in $\mathbb{R}^{3}$ . It follows that $\textbf{u} \times \textbf{v} \in \mathbb{R}^{3}$ and $\textbf{w} \times \textbf{x} \in \mathbb{R}^{3}$ ; it further follows that $(\textbf{u} \times \textbf{v}) \times( \textbf{w} \times \textbf{x}) \in \mathbb{R}^{3}$ .

Report an Error

Example Question #34 : Vector Vector Product

$\textbf{u},\textbf{v} \in \mathbb{R}^{3}$ ; $\textbf{w}, \textbf{x} \in \mathbb{R}^{4}$

Which of the following applies to $(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x})$ , where " $\times$ " refers to the cross product of two vectors, and "" refers to either scalar or vector addition, as applicable?

Possible Answers:

$(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x})$ is an undefined expression.

$(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x}) \in \mathbb{R}^{3}$

$(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x}) \in \mathbb{R}$

$(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x}) \in \mathbb{R}^{2}$

$(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x}) \in M_{2 \times 3}$

Correct answer:

$(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x})$ is an undefined expression.

Explanation:

The cross product of two vectors is defined only if both vectors are in $\mathbb{R}^{3}$ . $\textbf{w}$ and $\textbf{x}$ are vectors in $\mathbb{R}^{4}$ , so $\textbf{w} \times \textbf{x}$ is undefined; consequently, so is $(\textbf{u} \times \textbf{v}) + ( \textbf{w} \times \textbf{x})$ .

Report an Error

Example Question #35 : Vector Vector Product

$\textbf{u},\textbf{v} \in \mathbb{R}^{3}$ ; $\textbf{w}, \textbf{x} \in \mathbb{R}^{4}$

Which of the following applies to $(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x})$ , where " $\cdot$ " refers to the dot product of two vectors, and "" refers to either scalar or vector addition, as applicable?

Possible Answers:

$(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x})$ is an undefined expression.

$(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x}) \in M_{3 \times 4 }$

$(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x}) \in \mathbb{R}^{4}$

$(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x}) \in \mathbb{R}^{3}$

$(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x}) \in \mathbb{R}$

Correct answer:

$(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x}) \in \mathbb{R}$

Explanation:

The dot product of two vectors in the same vector space is a scalar quantity. $\textbf{u},\textbf{v} \in \mathbb{R}^{3}$ , so $\textbf{u}$ and $\textbf{v}$ are in the same vector space; their dot product is defined, and $\textbf{u} \cdot \textbf{v} \in \mathbb{R}$ . For similar reasons, $\textbf{w} \cdot \textbf{x} \in \mathbb{R}$ . Therefore, their sum is defined, and $(\textbf{u} \cdot \textbf{v}) + ( \textbf{w} \cdot \textbf{x}) \in \mathbb{R}$ .

Report an Error

Example Question #36 : Vector Vector Product

$\textbf{u},\textbf{v} \in \mathbb{R}^{3}$ .

Which of the following applies to $\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v}$ , where " $\cdot$ " and " $\times$ " refer to the dot product and the cross product of two vectors, and "" refers to either scalar or vector addition, as applicable?

Possible Answers:

$\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v} \in M_{1 \times 3 }$

$\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v} \in \mathbb{R}$

$\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v} \in M_{3 \times 1 }$

$\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v}$ is an undefined expression.

$\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v} \in \mathbb{R} ^{3}$

Correct answer:

$\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v}$ is an undefined expression.

Explanation:

The cross product $\textbf{u} \times \textbf{v}$ of two vectors in $\mathbb{R}^{3}$ is also a vector in $\mathbb{R}^{3}$ . The dot product $\textbf{u} \cdot \textbf{v}$ of two such vectors is a scalar. Since a vector and a scalar cannot be added, $\textbf{u} \times \textbf{v} + \textbf{u} \cdot \textbf{v}$ is an undefined expression.

Report an Error

← Previous 1 2 … 81 82 83 84 85 86 87 88 89 90 91 92 Next →

View Linear Algebra Tutors

Yoonsik
Certified Tutor

Seoul National University, Bachelor of Science, Physics. University of Pennsylvania, Doctor of Philosophy, Atomic and Molecul...

View Linear Algebra Tutors

Kurosh
Certified Tutor

University of tehran, Bachelor of Engineering, Electrical Engineering.

View Linear Algebra Tutors

Godswill
Certified Tutor

Rivers State University, Bachelor of Engineering, Mechanical Engineering. Texas Southern University, Master of Science, Compu...

All Linear Algebra Resources

4 Diagnostic Tests 108 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

ACT Tutors in Washington DC, French Tutors in Dallas Fort Worth, Algebra Tutors in New York City, Statistics Tutors in Dallas Fort Worth, English Tutors in Atlanta, Biology Tutors in Miami, English Tutors in Chicago, English Tutors in Seattle, SSAT Tutors in Philadelphia, Computer Science Tutors in Chicago

Popular Courses & Classes

MCAT Courses & Classes in New York City, ISEE Courses & Classes in New York City, SAT Courses & Classes in San Francisco-Bay Area, SAT Courses & Classes in Chicago, ACT Courses & Classes in San Francisco-Bay Area, SAT Courses & Classes in Phoenix, LSAT Courses & Classes in Philadelphia, MCAT Courses & Classes in Boston, ACT Courses & Classes in Houston, SSAT Courses & Classes in Washington DC

Popular Test Prep

SAT Test Prep in Denver, SSAT Test Prep in San Diego, ACT Test Prep in Boston, GMAT Test Prep in Seattle, MCAT Test Prep in Seattle, SAT Test Prep in Seattle, ACT Test Prep in Houston, GMAT Test Prep in Washington DC, GMAT Test Prep in Denver, ISEE Test Prep in Miami

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

Linear Algebra : Linear Algebra

Study concepts, example questions & explanations for Linear Algebra

All Linear Algebra Resources

Example Questions

Example Question #27 : Vector Vector Product

Example Question #28 : Vector Vector Product

Example Question #29 : Vector Vector Product

Example Question #30 : Vector Vector Product

Example Question #841 : Linear Algebra

Example Question #32 : Vector Vector Product

Example Question #33 : Vector Vector Product

Example Question #34 : Vector Vector Product

Example Question #35 : Vector Vector Product

Example Question #36 : Vector Vector Product

All Linear Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: