벡터와 공간 전체 문제

Let $\text{u},\text{v},\,\text{and}\,\text{w}$ be three-dimensional vectors and $c$ be a real number. Prove the following properties of the cross product.

a. $\text{u}\,\times \,\text{u}=0$ b. $\text{u}\,\times \,(\text{v}+\text{w})=(\text{u}\,\times \,\text{v})+(\text{u}\,\times \,\text{w})$ c. $c(\text{u}\,\times \,\text{v})=(c\text{u})\,\times \,\text{v}=\text{u}\,\times \,(c\text{v})$ d. $\text{u}\cdot (\text{u}\,\times \,\text{v})=0$

Show that vectors $\text{u}=\langle 1,0,-8\rangle ,$ $\text{v}=\langle 0,1,6\rangle ,$ and $\text{w}=\langle -1,9,3\rangle$ satisfy the following properties of the cross product.

a. $\text{u}\,\times \,\text{u}=0$ b. $\text{u}\,\times \,(\text{v}+\text{w})=(\text{u}\,\times \,\text{v})+(\text{u}\,\times \,\text{w})$ c. $c(\text{u}\,\times \,\text{v})=(c\text{u})\,\times \,\text{v}=\text{u}\,\times \,(c\text{v})$ d. $\text{u}\cdot (\text{u}\,\times \,\text{v})=0$

Nonzero vectors $\text{u},\text{v},\,\text{and}\,\text{w}$ are said to be linearly dependent if one of the vectors is a linear combination of the other two. For instance, there exist two nonzero real numbers $\alpha$ and $\beta$ such that $\text{w}=\alpha \text{u}+\beta \text{v}.$ Otherwise, the vectors are called linearly independent. Show that $\text{u},\text{v},\,\text{and}\,\text{w}$ are coplanar if and only if they are linear dependent.

Consider vectors $\text{u}=\langle 1,4,-7\rangle ,$ $\text{v}=\langle 2,-1,4\rangle ,$ $\text{w}=\langle 0,-9,18\rangle ,$ and $\text{p}=\langle 0,-9,17\rangle .$

a. Show that $\text{u},\text{v},\,\text{and}\,\text{w}$ are coplanar by using their triple scalar product b. Show that $\text{u},\text{v},\,\text{and}\,\text{w}$ are coplanar, using the definition that there exist two nonzero real numbers $\alpha$ and $\beta$ such that $\text{w}=\alpha \text{u}+\beta \text{v}.$ c. Show that $\text{u},\text{v},\,\text{and}\,\text{p}$ are linearly independent—that is, none of the vectors is a linear combination of the other two.

Consider points $A(0,0,2),$ $B(1,0,2),$ $C(1,1,2),$ and $D(0,1,2).$ Are vectors $\vec{AB},$ $\vec{AC},$ and $\vec{AD}$ linearly dependent (that is, one of the vectors is a linear combination of the other two)?

Show that vectors $\text{i}+\text{j},$ $\text{i}-\text{j},$ and $\text{i}+\text{j}+\text{k}$ are linearly independent—that is, there do not exist two nonzero real numbers $\alpha$ and $\beta$ such that $\text{i}+\text{j}+\text{k}=\alpha (\text{i}+\text{j})+\beta (\text{i}-\text{j}).$

Let $\text{u}=\langle {u}_{1},{u}_{2}\rangle$ and $\text{v}=\langle {v}_{1},{v}_{2}\rangle$ be two-dimensional vectors. The cross product of vectors $\text{u}$ and $\text{v}$ is not defined. However, if the vectors are regarded as the three-dimensional vectors $\tilde{\text{u}}=\langle {u}_{1},{u}_{2},0\rangle$ and $\tilde{\text{v}}=\langle {v}_{1},{v}_{2},0\rangle ,$ respectively, then, in this case, we can define the cross product of $\tilde{\text{u}}$ and $\tilde{\text{v}}.$ In particular, in determinant notation, the cross product of $\tilde{\text{u}}$ and $\tilde{\text{v}}$ is given by

$$\tilde{\text{u}}\,\times \,\tilde{\text{v}}=|\begin{array}{ccc}\text{i} & \text{j} & \text{k} \\ {u}_{1} & {u}_{2} & 0 \\ {v}_{1} & {v}_{2} & 0\end{array}|.$$

Use this result to compute $(\text{i}\cos \,\theta +\text{j}\sin \,\theta )\,\times \,(\text{i}sin\theta -\text{j}cos\theta ),$ where $\theta$ is a real number.

Consider points $P(2,1),$ $Q(4,2),$ and $R(1,2).$

a. Find the area of triangle $P,Q,\,\text{and}\,R.$ b. Determine the distance from point $R$ to the line passing through $P\,\text{and}\,Q.$

Determine a vector of magnitude $10$ perpendicular to the plane passing through the x-axis and point $P(1,2,4).$

Determine a unit vector perpendicular to the plane passing through the z-axis and point $A(3,1,-2).$

Consider $\text{u}$ and $\text{v}$ two three-dimensional vectors. If the magnitude of the cross product vector $\text{u}\,\times \,\text{v}$ is $k$ times larger than the magnitude of vector $\text{u},$ show that the magnitude of $\text{v}$ is greater than or equal to $k,$ where $k$ is a natural number.

[T] Assume that the magnitudes of two nonzero vectors $\text{u}$ and $\text{v}$ are known. The function $f(\theta )=\| \text{u}\| \| \text{v}\| \sin \,\theta$ defines the magnitude of the cross product vector $\text{u}\,\times \,\text{v},$ where $\theta \in [0,\pi ]$ is the angle between $\text{u}\,\text{and}\,\text{v}.$

a. Graph the function $f.$ b. Find the absolute minimum and maximum of function $f.$ Interpret the results. c. If $\| \text{u}\| =5$ and $\| \text{v}\| =2,$ find the angle between $\text{u}\,\text{and}\,\text{v}$ if the magnitude of their cross product vector is equal to $9.$

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

a. Find the triple scalar product $\text{u}\cdot (\text{v}\,\times \,\text{w}).$ b. Find the volume of the parallelepiped with the adjacent edges $\text{u},\text{v},\text{and}\,\text{w}.$

Find all vectors $\text{w}=\langle {w}_{1},{w}_{2},{w}_{3}\rangle$ that satisfy the equation $\langle 1,1,1\rangle \,\times \,\text{w}=\langle -1,-1,2\rangle .$

Solve the equation $\text{w}\,\times \,\langle 1,0,-1\rangle =\langle 3,0,3\rangle ,$ where $\text{w}=\langle {w}_{1},{w}_{2},{w}_{3}\rangle$ 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 $30^\circ$ 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 $40^\circ$ 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 $\langle 0,1,-2\rangle$ 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 $35^\circ$ to produce a torque of $20$ ft-lbs?

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

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