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?

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

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

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

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

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}

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}

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]

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]

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]

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}

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}

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}

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}

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]

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]

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]

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

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.

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?

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

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}

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}

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}

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}

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.

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.

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?

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?

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.

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?

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.

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.

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

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