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Projection of F onto a Line

  • #1
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Homework Statement



The turnbuckle is tightened until the tension in the cable AB equals 2.9 kN. Determine the vector expression for the tension T as a force acting on member AD. Also find the magnitude of the projection of T along the line AC.


Homework Equations





The Attempt at a Solution



I was able to find the vector expression for T without any real trouble:

AB = <2.6, 1.7, -6.1>

|AB| = sqrt(2.62+ 1.72 + (-6.1)2)

T*AB/|AB| = 0.42364<2.6, 1.7, -6.1>

T = 1.101i + 0.7201j - 2.584k

I know this part is correct but I'm having difficulty doing the projection.

I tried taking taking the dot product of the T vector with the unit vector AC:

AC = ,2.6, -2.8, -6.1>

|AC| = sqrt(2.62 + (-2.8)2 + (-6.1)2)
|AC| = 7.1979

AC/|AC| = 0.3612, -0.389, 0.8475

T°AC/|AC| = 1.101*0.3612 +(-0.389*0.7201)+(-0.8475*-2.584)
= 2.2746

But it says my answer is wrong and I'm not sure where I'm going wrong.

Advice would be greatly appreciated.
 

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Answers and Replies

  • #2
Simon Bridge
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Isn't the z component of the unit vector AC/|AC| supposed to be negative?
 
  • #3
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Yes it is; it's a typo, but I did include the negative in the calculation below that.
 
  • #4
Simon Bridge
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Code:
octave:20> T=[1.101;0.7201;-2.584] [color=green]< the tension vector[/color]
T =

   1.10100
   0.72010
  -2.58400

octave:21> C=[2.6;-2.8;-6.1] [color=green]< this is AC vector[/color]
C =

   2.6000
  -2.8000
  -6.1000

octave:22> sqrt(C'*C) [color=green]< |AC| = √AC.AC[/color]
ans =  7.1979 [color=green]<---<<< agrees with your value[/color]

octave:23> C=C./sqrt(C'*C) [color=green]< set up unit vector[/color]
C =

   0.36122
  -0.38900
  -0.84747

octave:24> T'*C
ans =  2.3074 [color=green]<---<<< does not agree with yours[/color]
... it follows, unless I messed something up above, that you've made a mistake in your arithmetic someplace - go over it again, slowly.

I think your method is sound ... the projection ##p## of ##\vec{u}## on to ##\vec{v}## is $$p=\frac{\vec{u}\cdot\vec{v}}{|\vec{v}|}$$
 
Last edited:
  • #5
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I ran my calculations again. You answer is correct. I must have made a mistake when I typed in the numbers into my calculator. Thank you.
 
  • #6
Simon Bridge
Science Advisor
Homework Helper
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Those are the hardest mistakes to spot - especially if you make the same number-punch error repeatedly.
 

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