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Section exercises

Verbal

What are the characteristics of the letters that are commonly used to represent vectors?

lowercase, bold letter, usually u , v , w

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How is a vector more specific than a line segment?

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What are i and j , and what do they represent?

They are unit vectors. They are used to represent the horizontal and vertical components of a vector. They each have a magnitude of 1.

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When a unit vector is expressed as a , b , which letter is the coefficient of the i and which the j ?

The first number always represents the coefficient of the i , and the second represents the j .

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Algebraic

Given a vector with initial point ( 5 , 2 ) and terminal point ( 1 , 3 ) , find an equivalent vector whose initial point is ( 0 , 0 ) . Write the vector in component form a , b .

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Given a vector with initial point ( 4 , 2 ) and terminal point ( 3 , 3 ) , find an equivalent vector whose initial point is ( 0 , 0 ) . Write the vector in component form a , b .

7 , 5

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Given a vector with initial point ( 7 , 1 ) and terminal point ( 1 , 7 ) , find an equivalent vector whose initial point is ( 0 , 0 ) . Write the vector in component form a , b .

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For the following exercises, determine whether the two vectors u and v are equal, where u has an initial point P 1 and a terminal point P 2 and v has an initial point P 3 and a terminal point P 4 .

P 1 = ( 5 , 1 ) , P 2 = ( 3 , 2 ) , P 3 = ( 1 , 3 ) , and P 4 = ( 9 , 4 )

not equal

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P 1 = ( 2 , 3 ) , P 2 = ( 5 , 1 ) , P 3 = ( 6 , 1 ) , and P 4 = ( 9 , 3 )

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P 1 = ( 1 , 1 ) , P 2 = ( 4 , 5 ) , P 3 = ( 10 , 6 ) , and P 4 = ( 13 , 12 )

equal

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P 1 = ( 3 , 7 ) , P 2 = ( 2 , 1 ) , P 3 = ( 1 , 2 ) , and P 4 = ( 1 , 4 )

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P 1 = ( 8 , 3 ) , P 2 = ( 6 , 5 ) , P 3 = ( 11 , 8 ) , and P 4 = ( 9 , 10 )

equal

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Given initial point P 1 = ( 3 , 1 ) and terminal point P 2 = ( 5 , 2 ) , write the vector v in terms of i and j .

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Given initial point P 1 = ( 6 , 0 ) and terminal point P 2 = ( 1 , 3 ) , write the vector v in terms of i and j .

7 i 3 j

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For the following exercises, use the vectors u = i + 5 j , v = −2 i − 3 j ,  and w = 4 i j .

Find 4 v + 2 u

6 i 2 j

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For the following exercises, use the given vectors to compute u + v , u v , and 2 u − 3 v .

u = 2 , 3 , v = 1 , 5

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u = 3 , 4 , v = 2 , 1

u + v = 5 , 5 , u v = 1 , 3 , 2 u 3 v = 0 , 5

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Let v = −4 i + 3 j . Find a vector that is half the length and points in the same direction as v .

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Let v = 5 i + 2 j . Find a vector that is twice the length and points in the opposite direction as v .

10 i 4 j

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For the following exercises, find a unit vector in the same direction as the given vector.

b = −2 i + 5 j

2 29 29 i + 5 29 29 j

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d = 1 3 i + 5 2 j

2 229 229 i + 15 229 229 j

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u = −14 i + 2 j

7 2 10 i + 2 10 j

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For the following exercises, find the magnitude and direction of the vector, 0 θ < 2 π .

6 , 5

| v | = 7.810 , θ = 39.806 °

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−4 , −6

| v | = 7.211 , θ = 236.310°

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Given u = 3 i − 4 j and v = −2 i + 3 j , calculate u v .

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Given u = − i j and v = i + 5 j , calculate u v .

6

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Given u = 2 , 4 and v = 3 , 1 , calculate u v .

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Given u = 1 , 6 and v = 6 , 1 , calculate u v .

12

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Graphical

For the following exercises, given v , draw v , 3 v and 1 2 v .

For the following exercises, use the vectors shown to sketch u + v , u v , and 2 u .

For the following exercises, use the vectors shown to sketch 2 u + v .

For the following exercises, use the vectors shown to sketch u − 3 v .

For the following exercises, write the vector shown in component form.

Questions & Answers

the diagram of the digestive system
Assiatu Reply
How does twins formed
William Reply
They formed in two ways first when one sperm and one egg are splited by mitosis or two sperm and two eggs join together
Oluwatobi
what is genetics
Josephine Reply
Genetics is the study of heredity
Misack
how does twins formed?
Misack
What is manual
Hassan Reply
discuss biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles
Joseph Reply
what is biology
Yousuf Reply
the study of living organisms and their interactions with one another and their environments
AI-Robot
the study of living organisms and their interactions with one another and their environment.
Wine
discuss the biological phenomenon and provide pieces of evidence to show that it was responsible for the formation of eukaryotic organelles in an essay form
Joseph Reply
what is the blood cells
Shaker Reply
list any five characteristics of the blood cells
Shaker
lack electricity and its more savely than electronic microscope because its naturally by using of light
Abdullahi Reply
advantage of electronic microscope is easily and clearly while disadvantage is dangerous because its electronic. advantage of light microscope is savely and naturally by sun while disadvantage is not easily,means its not sharp and not clear
Abdullahi
cell theory state that every organisms composed of one or more cell,cell is the basic unit of life
Abdullahi
is like gone fail us
DENG
cells is the basic structure and functions of all living things
Ramadan
What is classification
ISCONT Reply
is organisms that are similar into groups called tara
Yamosa
in what situation (s) would be the use of a scanning electron microscope be ideal and why?
Kenna Reply
A scanning electron microscope (SEM) is ideal for situations requiring high-resolution imaging of surfaces. It is commonly used in materials science, biology, and geology to examine the topography and composition of samples at a nanoscale level. SEM is particularly useful for studying fine details,
Hilary
cell is the building block of life.
Condoleezza Reply
what is cell divisoin?
Aron Reply
Diversity of living thing
ISCONT
what is cell division
Aron Reply
Cell division is the process by which a single cell divides into two or more daughter cells. It is a fundamental process in all living organisms and is essential for growth, development, and reproduction. Cell division can occur through either mitosis or meiosis.
AI-Robot
What is life?
Allison Reply
life is defined as any system capable of performing functions such as eating, metabolizing,excreting,breathing,moving,Growing,reproducing,and responding to external stimuli.
Mohamed
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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