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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

it is the relatively stable flow of income
Chidubem Reply
what is circular flow of income
Divine Reply
branches of macroeconomics
SHEDRACK Reply
what is Flexible exchang rate?
poudel Reply
is gdp a reliable measurement of wealth
Atega Reply
introduction to econometrics
Husseini Reply
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Jahara
Good morning
Jorge
hi
abubakar
hi
Nmesoma
hi
Mahesh
Hi
Tom
Why is unemployment rate never zero at full employment?
Priyanka Reply
bcoz of existence of frictional unemployment in our economy.
Umashankar
what is flexible exchang rate?
poudel
due to existence of the pple with disabilities
Abdulraufu
the demand of a good rises, causing the demand for another good to fall
Rushawn Reply
is it possible to leave every good at the same level
Joseph
I don't think so. because check it, if the demand for chicken increases, people will no longer consume fish like they used to causing a fall in the demand for fish
Anuolu
is not really possible to let the value of a goods to be same at the same time.....
Salome
Suppose the inflation rate is 6%, does it mean that all the goods you purchase will cost 6% more than previous year? Provide with reasoning.
Geetha Reply
Not necessarily. To measure the inflation rate economists normally use an averaged price index of a basket of certain goods. So if you purchase goods included in the basket, you will notice that you pay 6% more, otherwise not necessarily.
Waeth
discus major problems of macroeconomics
Alii Reply
what is the problem of macroeconomics
Yoal
Economic growth Stable prices and low unemployment
Ephraim
explain inflationcause and itis degre
Miresa Reply
what is inflation
Getu
increase in general price levels
WEETO
Good day How do I calculate this question: C= 100+5yd G= 2000 T= 2000 I(planned)=200. Suppose the actual output is 3000. What is the level of planned expenditures at this level of output?
Chisomo Reply
how to calculate actual output?
Chisomo
how to calculate the equilibrium income
Beshir
Criteria for determining money supply
Thapase Reply
who we can define macroeconomics in one line
Muhammad
Aggregate demand
Mohammed
C=k100 +9y and i=k50.calculate the equilibrium level of output
Mercy Reply
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Amisha
money as unit of account means what?
Kalombe
A unit of account is something that can be used to value goods and services and make calculations
Jim
all of you please speak in English I can't understand you're language
Muhammad
I want to know how can we define macroeconomics in one line
Muhammad
it must be .9 or 0.9 no Mpc is greater than 1 Y=100+.9Y+50 Y-.9Y=150 0.1Y/0.1=150/0.1 Y=1500
Kalombe
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Kalombe
hi can someone help me on this question If a negative shocks shifts the IS curve to the left, what type of policy do you suggest so as to stabilize the level of output? discuss your answer using appropriate graph.
Galge Reply
if interest rate is increased this will will reduce the level of income shifting the curve to the left ◀️
Kalombe
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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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