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Find all vectors $\text{w}=\u27e8{w}_{1},{w}_{2},{w}_{3}\u27e9$ that satisfy the equation $\u27e81,1,1\u27e9\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\text{w}=\u27e8\mathrm{-1},\mathrm{-1},2\u27e9.$
$\text{w}=\u27e8{w}_{3}-1,{w}_{3}+1,{w}_{3}\u27e9,$ where ${w}_{3}$ is any real number
Solve the equation $\text{w}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\u27e81,0,\mathrm{-1}\u27e9=\u27e83,0,3\u27e9,$ where $\text{w}=\u27e8{w}_{1},{w}_{2},{w}_{3}\u27e9$ is a nonzero vector with a magnitude of $3.$
[T] A mechanic uses a 12-in. wrench to turn a bolt. The wrench makes a $30\text{\xb0}$ angle with the horizontal. If the mechanic applies a vertical force of $10$ lb on the wrench handle, what is the magnitude of the torque at point $P$ (see the following figure)? Express the answer in foot-pounds rounded to two decimal places.
8.66 ft-lb
[T] A boy applies the brakes on a bicycle by applying a downward force of $20$ lb on the pedal when the 6-in. crank makes a $40\text{\xb0}$ angle with the horizontal (see the following figure). Find the torque at point $P.$ Express your answer in foot-pounds rounded to two decimal places.
[T] Find the magnitude of the force that needs to be applied to the end of a 20-cm wrench located on the positive direction of the y -axis if the force is applied in the direction $\u27e80,1,\mathrm{-2}\u27e9$ and it produces a $100$ N·m torque to the bolt located at the origin.
250 N
[T] What is the magnitude of the force required to be applied to the end of a 1-ft wrench at an angle of $35\text{\xb0}$ to produce a torque of $20$ N·m?
[T] The force vector $\text{F}$ acting on a proton with an electric charge of $1.6\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-19}}\text{C}$ (in coulombs) moving in a magnetic field $\text{B}$ where the velocity vector $\text{v}$ is given by $\text{F}=1.6\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-19}}\left(\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\mathbf{\text{B}}\right)$ (here, $\text{v}$ is expressed in meters per second, $\text{B}$ is in tesla [T], and $\text{F}$ is in newtons [N]). Find the force that acts on a proton that moves in the xy -plane at velocity $\text{v}={10}^{5}\text{i}+{10}^{5}\text{j}$ (in meters per second) in a magnetic field given by $\mathbf{\text{B}}=0.3\text{j}.$
$\text{F}=4.8\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-15}}\text{k}\phantom{\rule{0.2em}{0ex}}\mathbf{\text{N}}$
[T] The force vector $\text{F}$ acting on a proton with an electric charge of $1.6\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-19}}\text{C}$ moving in a magnetic field $\text{B}$ where the velocity vector v is given by $\text{F}=1.6\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-19}}\left(\text{v}\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}\mathbf{\text{B}}\right)$ (here, $\text{v}$ is expressed in meters per second, $\text{B}$ in $\text{T},$ and $\text{F}$ in $\text{N}).$ If the magnitude of force $\text{F}$ acting on a proton is $5.9\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{\mathrm{-17}}$ N and the proton is moving at the speed of 300 m/sec in magnetic field $\text{B}$ of magnitude 2.4 T, find the angle between velocity vector $\text{v}$ of the proton and magnetic field $\text{B}.$ Express the answer in degrees rounded to the nearest integer.
[T] Consider $\mathbf{\text{r}}(t)=\u27e8\text{cos}\phantom{\rule{0.2em}{0ex}}t,\text{sin}\phantom{\rule{0.2em}{0ex}}t,2t\u27e9$ the position vector of a particle at time $t\in [0,30],$ where the components of $\text{r}$ are expressed in centimeters and time in seconds. Let $\overrightarrow{OP}$ be the position vector of the particle after $1$ sec.
a.
$\mathbf{\text{B}}(t)=\u27e8\frac{2\phantom{\rule{0.2em}{0ex}}\text{sin}\phantom{\rule{0.2em}{0ex}}t}{\sqrt{5}},-\frac{2\phantom{\rule{0.2em}{0ex}}\text{cos}\phantom{\rule{0.2em}{0ex}}t}{\sqrt{5}},\frac{1}{\sqrt{5}}\u27e9;$
b.
A solar panel is mounted on the roof of a house. The panel may be regarded as positioned at the points of coordinates (in meters) $A(8,0,0),$ $B(8,18,0),$ $C(0,18,8),$ and $D(0,0,8)$ (see the following figure).
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