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a = 2 u + v w , where u = i k , v = 2 j , and w = i j

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Determine whether A B and P Q are equivalent vectors, where A ( 1 , 1 , 1 ) , B ( 3 , 3 , 3 ) , P ( 1 , 4 , 5 ) , and Q ( 3 , 6 , 7 ) .

Equivalent vectors

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Determine whether the vectors A B and P Q are equivalent, where A ( 1 , 4 , 1 ) , B ( −2 , 2 , 0 ) , P ( 2 , 5 , 7 ) , and Q ( −3 , 2 , 1 ) .

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For the following exercises, find vector u with a magnitude that is given and satisfies the given conditions.

v = 7 , −1 , 3 , u = 10 , u and v have the same direction

u = 70 59 , 10 59 , 30 59

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v = 2 , 4 , 1 , u = 15 , u and v have the same direction

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v = 2 sin t , 2 cos t , 1 , u = 2 , u and v have opposite directions for any t , where t is a real number

u = 4 5 sin t , 4 5 cos t , 2 5

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v = 3 sinh t , 0 , 3 , u = 5 , u and v have opposite directions for any t , where t is a real number

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Determine a vector of magnitude 5 in the direction of vector A B , where A ( 2 , 1 , 5 ) and B ( 3 , 4 , −7 ) .

5 154 , 15 154 , 60 154

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Find a vector of magnitude 2 that points in the opposite direction than vector A B , where A ( −1 , −1 , 1 ) and B ( 0 , 1 , 1 ) . Express the answer in component form.

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Consider the points A ( 2 , α , 0 ) , B ( 0 , 1 , β ) , and C ( 1 , 1 , β ) , where α and β are negative real numbers. Find α and β such that O A O B + O C = O B = 4 .

α = 7 , β = 15

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Consider points A ( α , 0 , 0 ) , B ( 0 , β , 0 ) , and C ( α , β , β ) , where α and β are positive real numbers. Find α and β such that O A + O B = 2 and O C = 3 .

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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 −2 x + 3 y + z = 2 .

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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 8 x + 2 y + 4 z = 21 .

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The points A , B , and C are collinear (in this order) if the relation A B + B C = A C is satisfied. Show that A ( 5 , 3 , −1 ) , B ( −5 , −3 , 1 ) , and C ( −15 , −9 , 3 ) are collinear points.

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

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[T] A force F of 50 N acts on a particle in the direction of the vector O P , where P ( 3 , 4 , 0 ) .

  1. Express the force as a vector in component form.
  2. Find the angle between force F and the positive direction of the x -axis. Express the answer in degrees rounded to the nearest integer.

a. F = 30 , 40 , 0 ; b. 53 °

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[T] A force F of 40 N acts on a box in the direction of the vector O P , where P ( 1 , 0 , 2 ) .

  1. Express the force as a vector by using standard unit vectors.
  2. Find the angle between force F and the positive direction of the x -axis.
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If 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 D = ( x 2 x 1 ) i + ( y 2 y 1 ) j + ( z 2 z 1 ) k . A metal container is lifted 10 m vertically by a constant force F . Express the displacement vector D by using standard unit vectors.

D = 10 k

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

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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 F 1 = 10 , 6 , 3 , F 2 = 0 , 4 , 9 , and F 3 = 10 , −3 , −9 be three forces acting on a box. Find the force F 4 acting on the box such that the box is in static equilibrium. Express the answer in component form.

F 4 = −20 , −7 , −3

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[T] Let F k = 1 , k , k 2 , k = 1 ,... , n be n forces acting on a particle, with n 2 .

  1. Find the net force F = k = 1 n F k . Express the answer using standard unit vectors.
  2. Use a computer algebra system (CAS) to find n such that F < 100 .
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The force of gravity F acting on an object is given by F = m g , where m is the mass of the object (expressed in kilograms) and g is acceleration resulting from gravity, with g = 9.8 N/kg . A 2-kg disco ball hangs by a chain from the ceiling of a room.

  1. Find the force of gravity F acting on the disco ball and find its magnitude.
  2. Find the force of tension T in the chain and its magnitude.
    Express the answers using standard unit vectors.
This figure shows a disco ball suspended from a ceiling.
(credit: modification of work by Kenneth Lu, Flickr)

a. F = −19.6 k , F = 19.6 N; b. T = 19.6 k , T = 19.6 N

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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.

  1. Find the magnitude of the force of gravity acting on the chandelier.
  2. Find the magnitudes of the forces of tension for each of the four chains (assume chains are essentially vertical).
This figure shows a light fixture hung from a ceiling, supported by 4 chains from the same point on the ceiling to four points spread evenly around the light fixture.
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[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 ( 1 , 3 , 0 ) , and R ( 1 , 3 , 0 ) . The load is located at S ( 0 , 0 , −2 3 ) , as shown in the following figure. Let F 1 , F 2 , and F 3 be the forces of tension resulting from the load in cables R S , Q S , and P S , respectively.

  1. Find the gravitational force F acting on the block of cement that counterbalances the sum F 1 + F 2 + F 3 of the forces of tension in the cables.
  2. Find forces F 1 , F 2 , and F 3 . Express the answer in component form.
This figure is the 3-dimensional coordinate system. It has 4 points drawn. The first point is labeled “P(-2, 0, 0).” The second point is labeled “R(1, -squareroot of 3, 0).” The third point is labeled “S(0, 0, -2squareroots of 3).” The fourth point is labeled “Q(1, squareroot of 3, 0).” There are line segments from P to S, from R to S, and from Q to S. At point S there is a box labeled “30 k g.”

a. F = −294 k N; b. F 1 = 49 3 3 , 49 , −98 , F 2 = 49 3 3 , −49 , −98 , and F 3 = 98 3 3 , 0 , −98 (each component is expressed in newtons)

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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 ° 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 ° to her teammate, who is located at point A . Write the velocity of the ball in component form.

This figure is the image of two soccer players. The first soccer player is at point A. The second player is at point C. There is a line segment from A to C. Ther is a vector from player C upwards labeled “v.” There is a vector from player A to the bottom of the image. The point at the bottom is labeled “B.” This vector is labeled “10m.” There is a vector from C to B labeled “10m.”
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Let r ( t ) = x ( t ) , y ( t ) , z ( t ) be the position vector of a particle at the time t [ 0 , T ] , where x , y , and z are smooth functions on [ 0 , T ] . The instantaneous velocity of the particle at time t is defined by vector v ( t ) = x ( t ) , y ( t ) , z ( t ) , with components that are the derivatives with respect to t , of the functions x , y , and z , respectively. The magnitude v ( t ) of the instantaneous velocity vector is called the speed of the particle at time t. Vector a ( t ) = x ( t ) , y ( t ) , z ( t ) , with components that are the second derivatives with respect to t , of the functions x , y , and z , respectively, gives the acceleration of the particle at time t . Consider r ( t ) = cos t , sin t , 2 t the position vector of a particle at time t [ 0 , 30 ] , where the components of r are expressed in centimeters and time is expressed in seconds.

  1. Find the instantaneous velocity, speed, and acceleration of the particle after the first second. Round your answer to two decimal places.
  2. Use a CAS to visualize the path of the particle—that is, the set of all points of coordinates ( cos t , sin t , 2 t ) , where t [ 0 , 30 ] .

a. v ( 1 ) = −0.84 , 0.54 , 2 (each component is expressed in centimeters per second); v ( 1 ) = 2.24 (expressed in centimeters per second); a ( 1 ) = −0.54 , −0.84 , 0 (each component expressed in centimeters per second squared);

b.
This figure is of the 3-dimensional coordinate system above the xy-plane. It has a spiral drawn resembling a spring. The spiral is around the z-axis. The spiral starts on the x-axis at x = 1.

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[T] Let r ( t ) = t , 2 t 2 , 4 t 2 be the position vector of a particle at time t (in seconds), where t [ 0 , 10 ] (here the components of r are expressed in centimeters).

  1. Find the instantaneous velocity, speed, and acceleration of the particle after the first two seconds. Round your answer to two decimal places.
  2. Use a CAS to visualize the path of the particle defined by the points ( t , 2 t 2 , 4 t 2 ) , where t [ 0 , 60 ] .
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Questions & Answers

Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
can you provide the details of the parametric equations for the lines that defince doubly-ruled surfeces (huperbolids of one sheet and hyperbolic paraboloid). Can you explain each of the variables in the equations?
Radek Reply
Practice Key Terms 6

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Source:  OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2
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