What is the vector expression for tension T and its projection along line AC?

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Discussion Overview

The discussion revolves around determining the vector expression for tension T in a cable system and calculating its projection along a specified line AC. The context is a homework problem involving vector mathematics and projections in a physics or engineering scenario.

Discussion Character

  • Homework-related
  • Mathematical reasoning

Main Points Raised

  • One participant presents their calculation for the tension vector T as T = 1.101i + 0.7201j - 2.584k and expresses confidence in this result.
  • The same participant struggles with calculating the projection of T along line AC and provides their method involving the dot product.
  • Another participant questions the sign of the z component in the unit vector for AC, suggesting a potential error.
  • A subsequent reply confirms that the z component was indeed a typo but asserts that it was corrected in their calculations.
  • Further calculations are shared using Octave, showing agreement on the magnitude of AC but a discrepancy in the projection result, leading to a suggestion that the original poster may have made an arithmetic error.
  • The original poster acknowledges a possible mistake in their calculations after re-evaluating their work and thanks the participant for their assistance.
  • Another participant comments on the difficulty of spotting repeated numerical errors in calculations.

Areas of Agreement / Disagreement

Participants generally agree on the method for calculating the tension vector and the projection, but there is a discrepancy in the numerical results initially presented by the original poster. The discussion reflects a collaborative effort to identify and correct errors without reaching a definitive conclusion on the projection value.

Contextual Notes

There are unresolved aspects regarding the accuracy of the arithmetic calculations and the potential for typographical errors in the initial setup of the problem. The discussion does not clarify the final correct projection value.

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

   1.10100
   0.72010
  -2.58400

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

   2.6000
  -2.8000
  -6.1000

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

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

   0.36122
  -0.38900
  -0.84747

octave:24> T'*C
ans =  2.3074 <---<<< 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:
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.
 
Those are the hardest mistakes to spot - especially if you make the same number-punch error repeatedly.
 

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