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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.

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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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