벡터와 공간 전체 문제

$\text{v}=\langle 2\,\sin \,t,2\,\cos \,t,1\rangle ,$ $\| \text{u}\| =2,$ $\text{u}$ and $\text{v}$ have opposite directions for any $t,$ where $t$ is a real number

$\text{v}=\langle 3\,\sinh \,t,0,3\rangle ,$ $\| \text{u}\| =5,$ $\text{u}$ and $\text{v}$ have opposite directions for any $t,$ where $t$ is a real number

Determine a vector of magnitude $5$ in the direction of vector $\vec{AB},$ where $A(2,1,5)$ and $B(3,4,-7).$

Find a vector of magnitude $2$ that points in the opposite direction than vector $\vec{AB},$ where $A(-1,-1,1)$ and $B(0,1,1).$ Express the answer in component form.

Consider the points $A(2,\alpha ,0),B(0,1,\beta ),$ and $C(1,1,\beta ),$ where $\alpha$ and $\beta$ are negative real numbers. Find $\alpha$ and $\beta$ such that $\| \vec{OA}-\vec{OB}+\vec{OC}\| =\| \vec{OB}\| =4.$

Consider points $A(\alpha ,0,0),B(0,\beta ,0),$ and $C(\alpha ,\beta ,\beta ),$ where $\alpha$ and $\beta$ are positive real numbers. Find $\alpha$ and $\beta$ such that $\| \overset{-}{OA}+\overset{-}{OB}\| =\sqrt{2}\,\text{and}\,\| \overset{-}{OC}\| =\sqrt{3}.$

Let $P(x,y,z)$ be a point situated at an equal distance from points $A(1,-1,0)$ and $B(-1,2,1).$ Show that point $P$ lies on the plane of equation $-2x+3y+z=2.$

Let $P(x,y,z)$ be a point situated at an equal distance from the origin and point $A(4,1,2).$ Show that the coordinates of point $P$ satisfy the equation $8x+2y+4z=21.$

The points $A,B,$ and $C$ are collinear (in this order) if the relation $\| \vec{AB}\| +\| \vec{BC}\| =\| \vec{AC}\|$ is satisfied. Show that $A(5,3,-1),$ $B(-5,-3,1),$ and $C(-15,-9,3)$ are collinear points.

For the following exercises, find vector $\text{u}$ with a magnitude that is given and satisfies the given conditions.

Show that points $A(1,0,1),$ $B(0,1,1),$ and $C(1,1,1)$ are not collinear.

[T] A force $\text{F}$ of $50\,\text{N}$ acts on a particle in the direction of the vector $\vec{OP},$ where $P(3,4,0).$

a. Express the force as a vector in component form. b. Find the angle between force $\text{F}$ and the positive direction of the x-axis. Express the answer in degrees rounded to the nearest integer.

[T] A force $\text{F}$ of $40\,\text{N}$ acts on a box in the direction of the vector $\vec{OP},$ where $P(1,0,2).$

a. Express the force as a vector by using standard unit vectors. b. Find the angle between force $\text{F}$ and the positive direction of the x-axis.

If $\text{F}$ is a force that moves an object from point ${P}_{1}({x}_{1},{y}_{1},{z}_{1})$ to another point ${P}_{2}({x}_{2},{y}_{2},{z}_{2}),$ then the displacement vector is defined as $\text{D}=({x}_{2}-{x}_{1})\text{i}+({y}_{2}-{y}_{1})\text{j}+({z}_{2}-{z}_{1})\text{k}.$ A metal container is lifted $10$ m vertically by a constant force $\text{F}.$ Express the displacement vector $\text{D}$ by using standard unit vectors.

A box is pulled $4$ yd horizontally in the x-direction by a constant force $\text{F}.$ Find the displacement vector in component form.

The sum of the forces acting on an object is called the resultant or net force. An object is said to be in static equilibrium if the resultant force of the forces that act on it is zero. Let ${\text{F}}_{1}=\langle 10,6,3\rangle ,$ ${\text{F}}_{2}=\langle 0,4,9\rangle ,$ and ${\text{F}}_{3}=\langle 10,-3,-9\rangle$ be three forces acting on a box. Find the force ${\text{F}}_{4}$ acting on the box such that the box is in static equilibrium. Express the answer in component form.

[T] Let ${\text{F}}_{k}=\langle 1,k,{k}^{2}\rangle ,$ $k=1,...,n$ be $n$ forces acting on a particle, with $n\ge 2.$

a. Find the net force $\text{F}=\sum_{k=1}^{n}{F}_{k}.$ Express the answer using standard unit vectors. b. Use a computer algebra system (CAS) to find n such that $\| \text{F}\| <100.$

The force of gravity $\text{F}$ acting on an object is given by $\text{F}=m\text{g},$ where m is the mass of the object (expressed in kilograms) and $\text{g}$ is acceleration resulting from gravity, with $\| \text{g}\| =9.8$ $\text{N/kg}.$ A 2-kg disco ball hangs by a chain from the ceiling of a room.

a. Find the force of gravity $\text{F}$ acting on the disco ball and find its magnitude. b. Find the force of tension $\text{T}$ in the chain and its magnitude. Express the answers using standard unit vectors.

(credit: modification of work by Kenneth Lu, Flickr)

A 5-kg pendant chandelier is designed such that the alabaster bowl is held by four chains of equal length, as shown in the following figure.

a. Find the magnitude of the force of gravity acting on the chandelier. b. Find the magnitudes of the forces of tension for each of the four chains (assume chains are essentially vertical).

[T] A 30-kg block of cement is suspended by three cables of equal length that are anchored at points $P(-2,0,0),$ $Q\left(1,\sqrt{3},0\right),$ and $R\left(1,-\sqrt{3},0\right).$ The load is located at $S\left(0,0,-2\sqrt{3}\right),$ as shown in the following figure. Let ${\text{F}}_{1},$ ${\text{F}}_{2},$ and ${\text{F}}_{3}$ be the forces of tension resulting from the load in cables $RS,QS,$ and $PS,$ respectively.

a. Find the gravitational force $\text{F}$ acting on the block of cement that counterbalances the sum ${\text{F}}_{1}+{\text{F}}_{2}+{\text{F}}_{3}$ of the forces of tension in the cables. b. Find forces ${\text{F}}_{1},$ ${\text{F}}_{2},$ and ${\text{F}}_{3}.$ Express the answer in component form.

Two soccer players are practicing for an upcoming game. One of them runs 10 m from point A to point B. She then turns left at $90^\circ$ and runs 10 m until she reaches point C. Then she kicks the ball with a speed of 10 m/sec at an upward angle of $45^\circ$ to her teammate, who is located at point A. Write the velocity of the ball in component form.