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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

preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply
hil
Roy Reply
hi
Roy Reply
A bridge is to be built in the shape of a semi-elliptical arch and is to have a span of 120 feet. The height of the arch at a distance of 40 feet from the center is to be 8 feet. Find the height of the arch at its center
Abdulfatah Reply

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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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