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

[Northbysouth]

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.

 

[Simon Bridge]

Isn't the z component of the unit vector AC/|AC| supposed to be negative?

 

[Northbysouth]

Yes it is; it's a typo, but I did include the negative in the calculation below that.

 

[Simon Bridge]

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 $$p=\frac{\vec{u}\cdot\vec{v}}{|\vec{v}|}$$

 

[Northbysouth]

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.

 

[Simon Bridge]

Those are the hardest mistakes to spot - especially if you make the same number-punch error repeatedly.