7.7 Chapter Review
Chapter
Chapter 7
Section
7.7
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} .

Q1a

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

\displaystyle -5 \vec{u}

Q2a

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

Q2b

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

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.

Q3

Calculate the dot product of each pair of vectors. Round your answers to two decimal places. Q4a

Calculate the dot product of each pair of vectors. Round your answers to two decimal places. Q4b

Calculate the dot product of each pair of vectors.

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

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}

Q5b

Calculate the dot product of each pair of vectors.

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

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]

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.

Q7

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

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

Q8a

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

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

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}

Q9a

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

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

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]

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]

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).

Q11

Determine the exact magnitude of each vector.

\displaystyle \overrightarrow{\mathrm{AB}} 

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

Q12a

Determine the exact magnitude of each vector.

\displaystyle \overrightarrow{\mathrm{PQ}} 

joining

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

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} 

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} 

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}) 

Q13c

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

Q14

Determine  \vec{u} \times \vec{v}  for each pair of vectors. Q15a

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

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

Q15b

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

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

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} 

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? Determine the projection of \vec{u} on \vec{v}, and its magnitude,
\displaystyle \vec{u}=[-2,5,3]\\\vec{v}=[4,-8,9]