내적과 정사영

For the following problems, the vector $\text{u}$ is given.

a. Find the direction cosines for the vector $\text{u}$. b. Find the direction angles for the vector $\text{u}$ expressed in degrees. (Round the answer to the nearest integer.)

$\text{u}=\langle -1,5,2\rangle$

For the following problems, the vector $\text{u}$ is given.

a. Find the direction cosines for the vector $\text{u}$. b. Find the direction angles for the vector $\text{u}$ expressed in degrees. (Round the answer to the nearest integer.)

$\text{u}=\langle 2,3,4\rangle$

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

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

For the following exercises, the vectors $\text{u}$ and $\text{v}$ are given.

a. Find the vector projection $\text{w}={\text{proj}}_{\text{u}}\text{v}$ of vector $\text{v}$ onto vector $\text{u}$. Express your answer in component form. b. Find the scalar projection ${\text{comp}}_{\text{u}}\text{v}$ of vector $\text{v}$ onto vector u.

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

For the following exercises, the vectors $\text{u}$ and $\text{v}$ are given.

a. Find the vector projection $\text{w}={\text{proj}}_{\text{u}}\text{v}$ of vector $\text{v}$ onto vector $\text{u}$. Express your answer in component form. b. Find the scalar projection ${\text{comp}}_{\text{u}}\text{v}$ of vector $\text{v}$ onto vector u.

$\text{u}=\langle -4,7\rangle ,$ $\text{v}=\langle 3,5\rangle$

For the following exercises, the vectors $\text{u}$ and $\text{v}$ are given.

a. Find the vector projection $\text{w}={\text{proj}}_{\text{u}}\text{v}$ of vector $\text{v}$ onto vector $\text{u}$. Express your answer in component form. b. Find the scalar projection ${\text{comp}}_{\text{u}}\text{v}$ of vector $\text{v}$ onto vector u.

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

For the following exercises, the vectors $\text{u}$ and $\text{v}$ are given.

a. Find the vector projection $\text{w}={\text{proj}}_{\text{u}}\text{v}$ of vector $\text{v}$ onto vector $\text{u}$. Express your answer in component form. b. Find the scalar projection ${\text{comp}}_{\text{u}}\text{v}$ of vector $\text{v}$ onto vector u.

$\text{u}=\langle 4,4,0\rangle ,$ $\text{v}=\langle 0,4,1\rangle$

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

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

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

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

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

a. Find the distance between the hydrogen atoms located at P and R. b. Find the angle between vectors $\vec{OS}$ and $\vec{OR}$ 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 $\text{F}=\langle 5,6,-2\rangle$ (measured in Newtons) that moves a particle from point $P(3,-1,0)$ to point $Q(2,3,1)$ 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 $25^\circ$ 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 $20^\circ$ 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 $(1\text{J}=1\text{N}\cdot \text{m})$ rounded to the nearest integer.

[T] A boat sails north aided by a wind blowing in a direction of $\text{N3}0^\circ \text{E}$ 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 $\text{p}=\langle 150,225,375\rangle$ represents the price of certain models of bicycles sold by a bicycle shop. Vector $\text{n}=\langle 10,7,9\rangle$ represents the number of bicycles sold of each model, respectively. Compute the dot product $\text{p}\cdot \text{n}$ 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 ${\text{F}}_{2}.$

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

[T] Consider $\text{r}(t)=\langle \cos \,t,\sin \,t,2t\rangle$ 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 $\vec{OP}$ be the position vector of the particle after 1 sec.

a. Show that all vectors $\vec{PQ},$ 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 $\vec{PQ}=\langle x-\cos \,1,y-\sin \,1,z-2\rangle ,$ where $x\,\sin \,1-y\,\cos \,1-2z+4=0.$ The set of point Q describes a plane called the normal plane to the path of the particle at point P. b. Use a CAS to visualize the instantaneous velocity vector and the normal plane at point P along with the path of the particle.