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

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

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

\displaystyle -5 \vec{u}

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

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

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.

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

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

Calculate the dot product of each pair of vectors.

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

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}

Calculate the dot product of each pair of vectors.

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

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]

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.

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

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

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

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

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}

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

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

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]

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]

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).

Determine the exact magnitude of each vector.

\displaystyle \overrightarrow{\mathrm{AB}}

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

Determine the exact magnitude of each vector.

\displaystyle \overrightarrow{\mathrm{PQ}}

joining

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

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}

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}

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})

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

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

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

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

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

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

Use an example to verify that

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

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]