2.4 The cross product  (Page 10/16)

Find all vectors $\text{w}=⟨w_1,w_2,w_3⟩$ that satisfy the equation $⟨1,1,1⟩\times \text{w}=⟨\mathrm{-1},\mathrm{-1},2⟩.$

Solve the equation $\text{w}\times ⟨1,0,\mathrm{-1}⟩=⟨3,0,3⟩,$ where $\text{w}=⟨w_1,w_2,w_3⟩$ 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 $30\text{°}$ 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.

[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 $40\text{°}$ 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 $⟨0,1,\mathrm{-2}⟩$ and it produces a $100$ N·m torque to the bolt located at the origin.

[T] What is the magnitude of the force required to be applied to the end of a 1-ft wrench at an angle of $35\text{°}$ to produce a torque of $20$ N·m?

[T] The force vector $\text{F}$ acting on a proton with an electric charge of $1.6\times {10}^{\mathrm{-19}}\text{C}$ (in coulombs) moving in a magnetic field $\text{B}$ where the velocity vector $\text{v}$ is given by $\text{F}=1.6\times {10}^{\mathrm{-19}}\left(\text{v}\times \mathbf{\text{B}}\right)$ (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 $\mathbf{\text{B}}=0.3\text{j}.$

[T] The force vector $\text{F}$ acting on a proton with an electric charge of $1.6\times {10}^{\mathrm{-19}}\text{C}$ moving in a magnetic field $\text{B}$ where the velocity vector v is given by $\text{F}=1.6\times {10}^{\mathrm{-19}}\left(\text{v}\times \mathbf{\text{B}}\right)$ (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 $5.9\times {10}^{\mathrm{-17}}$ 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 $\mathbf{\text{r}}(t)=⟨\text{cos}t,\text{sin}t,2t⟩$ 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.

1. Determine unit vector $\mathbf{\text{B}}(t)$ (called the binormal unit vector ) that has the direction of cross product vector $\text{v}(t)\times \mathbf{\text{a}}(t),$ where $\text{v}(t)$ and $\text{a}(t)$ are the instantaneous velocity vector and, respectively, the acceleration vector of the particle after $t$ seconds.
2. Use a CAS to visualize vectors $\text{v}(1),$ $\text{a}(1),$ and $\mathbf{\text{B}}(1)$ as vectors starting at point $P$ along with the path of the particle.

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).

1. Find vector $\mathbf{\text{n}}=\overrightarrow{AB}\times \overrightarrow{AD}$ perpendicular to the surface of the solar panels. Express the answer using standard unit vectors.
2. Assume unit vector $\mathbf{\text{s}}=\frac{1}{\sqrt{3}}\text{i}+\frac{1}{\sqrt{3}}\text{j}+\frac{1}{\sqrt{3}}\text{k}$ points toward the Sun at a particular time of the day and the flow of solar energy is $\text{F}=900\mathbf{\text{s}}$ (in watts per square meter [ ${\text{W/m}}^2$ ]). Find the predicted amount of electrical power the panel can produce, which is given by the dot product of vectors $\text{F}$ and $\text{n}$ (expressed in watts).
3. Determine the angle of elevation of the Sun above the solar panel. Express the answer in degrees rounded to the nearest whole number. ( Hint : The angle between vectors $\text{n}$ and $\text{s}$ and the angle of elevation are complementary.)