7.7 Chapter Review
Chapter
Chapter 7
Section
7.7
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Solutions 31 Videos

Consider the vector \displaystyle \vec{v}=[-6,3] .

Write \displaystyle \vec{v} in terms of \displaystyle \vec{i} and \displaystyle \vec{j} .

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Q1a

Given \displaystyle \vec{u}=[5,-2] and \displaystyle \vec{v}=[8,5] , evaluate each of the following.

\displaystyle -5 \vec{u}

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Q2a

Given \displaystyle \vec{u}=[5,-2] and \displaystyle \vec{v}=[8,5] , evaluate each of the following. \displaystyle \vec{u}+\vec{v}

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Q2b

\displaystyle 4 \vec{u}+2 \vec{v}

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Q2c

An airplane is flying at an airspeed of \displaystyle 345 \mathrm{~km} / \mathrm{h} on a heading of \displaystyle 040^{\circ} . The wind is blowing at \displaystyle 18 \mathrm{~km} / \mathrm{h} from a bearing of \displaystyle 087^{\circ} . Determine the ground velocity of the airplane. Include a diagram in your solution.

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Q3

Calculate the dot product of each pair of vectors. Round your answers to two decimal places.

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Q4a

Calculate the dot product of each pair of vectors. Round your answers to two decimal places.

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Q4b

Calculate the dot product of each pair of vectors.

\displaystyle \vec{u}=[5,2], \vec{v}=[-6,7]

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Q5a

Calculate the dot product of each pair of vectors.

\displaystyle \vec{u}=-3 \vec{i}+2 \vec{j}, \vec{v}=3 \vec{i}+7 \vec{j}

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Q5b

Calculate the dot product of each pair of vectors.

\displaystyle \vec{u}=[3,2], \vec{v}=[4,-6]

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Q5c

Which vectors from below are orthogonal? Explain.

\displaystyle \vec{u}=[5,2], \vec{v}=[-6,7] b) \displaystyle \vec{u}=-3 \vec{i}+2 \vec{j}, \vec{v}=3 \vec{i}+7 \vec{j} c) \displaystyle \vec{u}=[3,2], \vec{v}=[4,-6]

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Q6

Two vectors have magnitudes of \displaystyle 5.2 and \displaystyle 7.3 . The dot product of the vectors is \displaystyle 20 . What is the angle between the vectors? Round your answer to the nearest degree.

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Q7

Calculate the angle between the vectors in each pair. Illustrate geometrically.

\displaystyle \vec{a}=[6,-5], \vec{b}=[7,2]

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Q8a

Calculate the angle between the vectors in each pair. Illustrate geometrically.

\displaystyle \vec{p}=[-9,-4], \vec{q}=[7,-3]

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Q8b

Determine the projection of \displaystyle \vec{u} on \displaystyle \vec{\nu} .

\displaystyle |\vec{u}|=56,|\vec{v}|=100 , angle \displaystyle \theta between \displaystyle \vec{u} and \displaystyle \vec{v} is \displaystyle 125^{\circ}

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Q9a

Determine the projection of \vec{u} on \vec{v}.

\displaystyle \vec{u}=[7,1], \vec{v}=[9,-3]

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Q9b

Determine the work done by each force, \displaystyle \vec{F} , in newtons, for an object moving along the

vector \displaystyle \vec{d} , in metres. \displaystyle \vec{F}=[16,12], \vec{d}=[3,9]

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Q10a

Determine the work done by each force, \displaystyle \vec{F} , in newtons, for an object moving along the

vector \displaystyle \vec{d} , in metres.

\displaystyle \vec{F}=[200,2000], \vec{d}=[3,45]

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Q10b

An electronics store sells 40-GB digital music players for \$ 229 and 80 -GB players for \$ 329 . Last month, the store sold 125 of the 40 -GB players and 70 of the 80 -GB players.

a) Represent the total revenue from sales of the players using the dot product.

b) Find the total revenue in part a).

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Q11

Determine the exact magnitude of each vector.

\displaystyle \overrightarrow{\mathrm{AB}}

joining \displaystyle \mathrm{A}(2,7,8)\\\mathrm{B}(-5,9,-1)

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Q12a

Determine the exact magnitude of each vector.

\displaystyle \overrightarrow{\mathrm{PQ}}

joining

\displaystyle \mathrm{P}(0,3,6)\\\mathrm{Q}(4,-9,7)

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Q12b

Given the vectors \vec{a}=[3,-7,8], \vec{b}=[-6,3,4] , and \vec{c}=[2,5,7] , evaluate each expression.

\displaystyle 5 \vec{a}-4 \vec{b}+3 \vec{c}

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Q13a

Given the vectors \vec{a}=[3,-7,8], \vec{b}=[-6,3,4] , and \vec{c}=[2,5,7] , evaluate each expression.

\displaystyle -5 \vec{a} \cdot \vec{c}

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Q13b

Given the vectors \vec{a}=[3,-7,8], \vec{b}=[-6,3,4] , and \vec{c}=[2,5,7] , evaluate each expression.

\displaystyle \vec{b} \cdot(\vec{c}-\vec{a})

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Q13c

If \vec{u}=[6,1,8] is orthogonal to \vec{v}=[k,-4,5] , determine the value(s) of k .

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Q14

Determine \vec{u} \times \vec{v} for each pair of vectors.

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Q15a

Determine \vec{u} \times \vec{v} for each pair of vectors.

\displaystyle \vec{u}=[4,1,-3], \vec{v}=[3,7,8]

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Q15b

Determine the area of the parallelogram defined by the vectors \vec{u}=[6,8,9] and

\displaystyle \vec{v}=[3,-1,2]

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Q16

Use an example to verify that

\displaystyle \vec{a} \times(\vec{b}+\vec{c})=\vec{a} \times \vec{b}+\vec{a} \times \vec{c}

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Q17

A force of 200 \mathrm{~N} is

applied to a wrench in a clockwise

direction at 80^{\circ} to

the handle, 10 \mathrm{~cm} from the centre of

the bolt.

a) Calculate the

magnitude of the torque.

b) In what direction does the torque vector point?

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Q18

Determine the projection of \vec{u} on \vec{v}, and its magnitude,

\displaystyle \vec{u}=[-2,5,3]\\\vec{v}=[4,-8,9]

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Q19