2.3 The dot product  (Page 9/16)

$\text{u}=⟨2,2,1⟩$

$\text{u}=\text{i}-2\text{j}+2\text{k}$

$\text{u}=⟨\mathrm{-1},5,2⟩$

$\text{u}=⟨2,3,4⟩$

Consider $\text{u}=⟨a,b,c⟩$ a nonzero three-dimensional vector. Let $\text{cos}\alpha ,$ $\text{cos}\beta ,$ and $\text{cos}\gamma$ be the directions of the cosines of u . Show that ${\text{cos}}^2\alpha +{\text{cos}}^2\beta +{\text{cos}}^2\gamma =1.$

Determine the direction cosines of vector $\text{u}=\text{i}+2\text{j}+2\text{k}$ and show they satisfy ${\text{cos}}^2\alpha +{\text{cos}}^2\beta +{\text{cos}}^2\gamma =1.$

$\text{u}=5\text{i}+2\text{j},$ $\text{v}=2\text{i}+3\text{j}$

$\text{u}=⟨\mathrm{-4},7⟩,$ $\text{v}=⟨3,5⟩$

$\text{u}=3\text{i}+2\text{k},$ $\text{v}=2\mathbf{\text{j}}+4\text{k}$

$\text{u}=⟨4,4,0⟩,$ $\text{v}=⟨0,4,1⟩$

Consider the vectors $\text{u}=4\text{i}-3\text{j}$ and $\text{v}=3\text{i}+2\text{j}.$

1. Find the component form of vector $\text{w}={\text{proj}}_{\mathbf{\text{u}}}\mathbf{\text{v}}$ that represents the projection of v onto u .
2. Write the decomposition $\text{v}=\mathbf{\text{w}}+\mathbf{\text{q}}$ of vector v into the orthogonal components w and q , where w is the projection of v onto u and q is a vector orthogonal to the direction of u .

Consider vectors $\text{u}=2\text{i}+4\mathbf{\text{j}}$ and $\text{v}=4\mathbf{\text{j}}+2\text{k}.$

1. Find the component form of vector $\text{w}={\text{proj}}_{\text{u}}\mathbf{\text{v}}$ that represents the projection of v onto u .
2. Write the decomposition $\text{v}=\mathbf{\text{w}}+\mathbf{\text{q}}$ of vector v into the orthogonal components w and q , where w is the projection of v onto u and q is a vector orthogonal to the direction of u .

A methane molecule has a carbon atom situated at the origin and four hydrogen atoms located at points $P\left(1,1,\mathrm{-1}\right),Q\left(1,\mathrm{-1},1\right),R\left(\mathrm{-1},1,1\right),\text{and}S\left(\mathrm{-1},\mathrm{-1},\mathrm{-1}\right)$ (see figure).

1. Find the distance between the hydrogen atoms located at P and R .
2. Find the angle between vectors $\overrightarrow{OS}$ and $\overrightarrow{OR}$ that connect the carbon atom with the hydrogen atoms located at S and R , which is also called the bond angle . Express the answer in degrees rounded to two decimal places.

[T] Find the vectors that join the center of a clock to the hours 1:00, 2:00, and 3:00. Assume the clock is circular with a radius of 1 unit.

Find the work done by force $\text{F}=⟨5,6,\mathrm{-2}⟩$ (measured in Newtons) that moves a particle from point $P\left(3,\mathrm{-1},0\right)$ to point $Q\left(2,3,1\right)$ along a straight line (the distance is measured in meters).

[T] A sled is pulled by exerting a force of 100 N on a rope that makes an angle of $25\text{°}$ with the horizontal. Find the work done in pulling the sled 40 m. (Round the answer to one decimal place.)

[T] A father is pulling his son on a sled at an angle of $20\text{°}$ with the horizontal with a force of 25 lb (see the following image). He pulls the sled in a straight path of 50 ft. How much work was done by the man pulling the sled? (Round the answer to the nearest integer.)

[T] A car is towed using a force of 1600 N. The rope used to pull the car makes an angle of 25° with the horizontal. Find the work done in towing the car 2 km. Express the answer in joules $(1\text{J}=1\text{N}·\text{m})$ rounded to the nearest integer.

[T] A boat sails north aided by a wind blowing in a direction of $\text{N3}0\text{°}\text{E}$ with a magnitude of 500 lb. How much work is performed by the wind as the boat moves 100 ft? (Round the answer to two decimal places.)

Vector $\text{p}=⟨150,225,375⟩$ represents the price of certain models of bicycles sold by a bicycle shop. Vector $\text{n}=⟨10,7,9⟩$ represents the number of bicycles sold of each model, respectively. Compute the dot product $\text{p}·\text{n}$ and state its meaning.

[T] Two forces ${\text{F}}_1$ and ${\text{F}}_2$ are represented by vectors with initial points that are at the origin. The first force has a magnitude of 20 lb and the terminal point of the vector is point $P(1,1,0).$ The second force has a magnitude of 40 lb and the terminal point of its vector is point $Q(0,1,1).$ Let F be the resultant force of forces ${\text{F}}_1$ and ${\text{F}}_2.$

1. Find the magnitude of F . (Round the answer to one decimal place.)
2. Find the direction angles of F . (Express the answer in degrees rounded to one decimal place.)

[T] Consider $\mathbf{\text{r}}(t)=⟨\text{cos}t,\text{sin}t,2t⟩$ the position vector of a particle at time $t\in [0,30],$ where the components of r are expressed in centimeters and time in seconds. Let $\overrightarrow{OP}$ be the position vector of the particle after 1 sec.

1. Show that all vectors $\overrightarrow{PQ},$ where $Q(x,y,z)$ is an arbitrary point, orthogonal to the instantaneous velocity vector $\text{v}(1)$ of the particle after 1 sec, can be expressed as $\overrightarrow{PQ}=⟨x-\text{cos}1,y-\text{sin}1,z-2⟩,$ where $x\text{sin}1-y\text{cos}1-2z+4=0.$ The set of point Q describes a plane called the normal plane to the path of the particle at point P .
2. Use a CAS to visualize the instantaneous velocity vector and the normal plane at point P along with the path of the particle.