7.6 Applications of the Dot Product and Cross Product
Chapter
Chapter 7
Section
7.6
Solutions 34 Videos

A force of 90 N is applied to a wrench in a counterclockwise direction at 70^\circ to the handle, 15 cm from the centre of the bolt.

a) Calculate the magnitude of the torque.

b) In what direction does the bolt move?

1.08mins
Q1

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

\vec{u}=[3,1,4], \vec{v}=[6,2,7]

1.31mins
Q2a

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

\vec{u}=[5,-4,8], \vec{v}=[3,7,6]

1.11mins
Q2b

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

\vec{u} = -2\vec{i} - 7\vec{j} + 3\vec{k}, \vec{v} = 6\vec{i} + \vec{j} -8\vec{k}

Q2c

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

\vec{u} = \vec{i} - \vec{k}, \vec{v} = 9\vec{i} + \vec{j}

Q2d

A force, \vec{F}=[3,5,12], in newtons, is applied to lift a box, with displacement, \vec{s}, in metres as given. Calculate the work against gravity and compare it to the work in the direction of travel.

\vec{s}=[0,0,8]

0.18mins
Q3a

A force, \vec{F}=[3,5,12], in newtons, is applied to lift a box, with displacement, \vec{s}, in metres as given. Calculate the work against gravity and compare it to the work in the direction of travel.

\vec{s}=[2,0,10]

0.22mins
Q3b

A force, \vec{F}=[3,5,12], in newtons, is applied to lift a box, with displacement, \vec{s}, in metres as given. Calculate the work against gravity and compare it to the work in the direction of travel.

\vec{s}=[2,1,6]

0.24mins
Q3c

Given \vec{a}=[-2,3,5], \vec{b}=[4,0,-1], and \vec{c}=[2,-2,3], evaluate each expression.

\vec{a}\times\vec{b}\cdot\vec{c}

1.01mins
Q4a

Given \vec{a}=[-2,3,5], \vec{b}=[4,0,-1], and \vec{c}=[2,-2,3], evaluate each expression.

\vec{a}\cdot\vec{b}\times\vec{c}

1.28mins
Q4b

Given \vec{a}=[-2,3,5], \vec{b}=[4,0,-1], and \vec{c}=[2,-2,3], evaluate each expression.

\vec{a}\times\vec{c}\cdot\vec{b}

1.00mins
Q4c

Given \vec{a}=[-2,3,5], \vec{b}=[4,0,-1], and \vec{c}=[2,-2,3], evaluate each expression.

\vec{b}\cdot\vec{a}\times\vec{c}

0.29mins
Q4d

Find the volume of each parallelepiped, defined by the vectors \vec{u}, \vec{v}, and \vec{w}.

\vec{u}=[1,4,3], \vec{v}=[2,5,6], and \vec{w}=[1,2,7]

1.05mins
Q5a

Find the volume of each parallelepiped, defined by the vectors \vec{u}, \vec{v}, and \vec{w}.

\vec{u}=[-2,5,1], \vec{v}=[3,-4,2], and \vec{w}=[1,3,5]

1.23mins
Q5b

Find the volume of each parallelepiped, defined by the vectors \vec{u}, \vec{v}, and \vec{w}.

\vec{u}=[1,1,9], \vec{v}=[0,0,4], and \vec{w}=[-2,0,5]

0.44mins
Q5c

A triangle has vertices A(-2,1,3), B(7,8,-4), and C(5,0,2). Determine the area of \triangleABC.

2.31mins
Q6

A bicycle pedal is pushed by a 75-N force, exerted as shown in the diagram. The shaft of the pedal is 15 cm long. Find the magnitude of the torque vector, in newton-metres, about point A.

1.06mins
Q7

A 65-kg boy is sitting on a see saw 0.6 m from the balance point.How far from the balance point should a 40-kg girl sit so that the see saw remains balanced?

2.07mins
Q8

Given \vec{u}=[2,2,3], \vec{v}=[1,3,4], and \vec{w}=[6,2,1], evaluate each expression.

|\vec{u}\times\vec{v}|^2-(\vec{w}\cdot\vec{w})^2

1.52mins
Q9a

Given \vec{u}=[2,2,3], \vec{v}=[1,3,4], and \vec{w}=[6,2,1], evaluate each expression.

|\vec{u}\times\vec{u}|+\vec{u}\cdot\vec{u}

0.41mins
Q9b

Given \vec{u}=[2,2,3], \vec{v}=[1,3,4], and \vec{w}=[6,2,1], evaluate each expression.

\vec{u}\times\vec{v}\cdot\vec{v}\times\vec{w}

1.35mins
Q9c

Given \vec{u}=[2,2,3], \vec{v}=[1,3,4], and \vec{w}=[6,2,1], evaluate each expression.

\vec{u}\times\vec{v}\cdot\vec{u}\times\vec{w}

1.04mins
Q9d

Consider two vectors \vec{a} and \vec{b}.

In a single diagram, illustrate both |\vec{a}\times\vec{b}| and \vec{a}\cdot\vec{b}

0.44mins
Q10a

Consider two vectors \vec{a} and \vec{b}.

Interpret |\vec{a}\times\vec{b}|^2+|\vec{a}\cdot\vec{b}|^2 and illustrate it on your diagram.

0.38mins
Q10b

Consider two vectors \vec{a} and \vec{b}.

Show that |\vec{a}\times\vec{b}|^2+|\vec{a}\cdot\vec{b}|^2=|\vec{a}|^2|\vec{b}|^2.

0.43mins
Q10c

An axle has two wheels of radii 0.75 m and 0.35 m attached to it. A 10-N force is applied horizontally to the edge of the larger wheel and a 5-N weight hangs from the edge of the smaller wheel. What is the net torque acting on the axle?

2.19mins
Q11

When a wrench is rotated. the magnitude of the torque is 10 N\cdotm. An 80-N force is applied 20 cm from the fulcrum. At what angle to the wrench is the force applied?

1.06mins
Q12

Is the following statement true or false? "If \vec{e}\times\vec{f}=\vec{0}, then \vec{e}\cdot\vec{f}=\vec{0}." Justify your response.

0.25mins
Q14

A wrench is rotated with torque of magnitude 100 N\cdotm. The force is applied 30 cm from the fulcrum, at an angle of 40^\circ. What is the magnitude of the force, to one decimal place?

1.01mins
Q15

Three edges of a right triangular prism are defined by the vectors \vec{a}=[1,3,2], \vec{b}=[2,2,4], and \vec{c}=[12,0,-6].

a) Draw a diagram of the prism, identifying which edge of the prism is defined by each vector.

b) Determine the volume of the prism.

c) Explain how your method of solving this problem would change if the prism were not necessarily a right prism.

5.54mins
Q16

Given that \vec{w}=k\vec{u}+m\vec{v}, where k,m\in\mathbb{R}, prove algebraically that \vec{u}\times\vec{v}\cdot\vec{w}=0.

1.35mins
Q17

Prove that (\vec{a}\times\vec{b})\times\vec{c} is in the same plane as \vec{a} and \vec{b}.

1.54mins
Q18

Let \vec{u}, \vec{v} and \vec{w} be mutually orthogonal vectors. What can be said about \vec{u}+\vec{v}, \vec{u}+\vec{w}, and \vec{v}+\vec{w}

The figure shown is a regular decagon with side length 1 cm. Determine the exact value of x