Capability at geometry, as space is and how vectors span space.. ideas needed for thermodynamics. One not good at geometry. Will stink thermo. Be well-capable at geometry as ye can.

High Wire Apparatus A cubical steel frame measuring ten meters on each edge is erected in a stadium to support the high wire apparatus for performing dogs. A walking wire 0A is strung as shown. Wire BC is attached to the frame at B, eight meters from the floor. The other end, C, is attached H meters from the floor. H is chosen such that the wires contact at point, P. Confirm by calculation that the height, H, is two meters.

♦  Since the configuration of the wires is symmetric, one can deduce that the proper height is two meters. Calculations are needed to prove that fact. Dimensions and a vector basis
(I, J, K) are provided with the figure (all distances are meters). To begin, for H correctly chosen, the following implicit vector equality will be true.

0P = 0B + BP

Now write each term as explicitly as possible. Since we do not have the components, we represent 0P in a magnitude, unit vector form.

0P = |0P|e_0P

For the next vector, 0B, we look at the figure and see its components are known.

0B = 0I + 10J + 8K

BP is not known, write it in magnitude-direction form.

BP = |BP|e_BP

Putting these into our equation yields:

|0P|e_0P = (0 I + 10 J + 8 K) + |BP|e_BP

By a check, our steps are correct. There are four unknown entities in this equation. By observation of the figure we see these unknown unit vectors are equal to other unit vectors which can be written. Specifically:

e_0P = e_0A

e_BP = e_BC

It is apparent e_0A (the direction of the vector 0A) can be written. Lets do that!

0A = 10 I + 10 J + 10 K which has the length (or magnitude), |0A|:

|0A| = [10² + 10² + 10²]^½ = 10(3)^½

To complete this we multiply and divide 0A by its magnitude then arrange it to have e_0A.

Next we go after e_BC which is the direction of vector BC. We use the vector equation,

0B + BC = 0C.

Writing this directly,

( 0 I + 10 J + 8 K) + BC = (10 I + 0 J + H K)

which solves for BC and e_BC as:

Returning to the beginning equation, we have:

|0P|e_0A = ( 0 I + 10 J + 8 K ) + |BP|e_BC

So we put the pieces into it.

Vectors are ordered numbers associated with previously prescribed directions. The ter, "ordered triple" is used sometimes. The vector procedure is always the same. The next task is to separate the above messy equation (carefully) into components. One way is to collect all I, J and K pieces left of equality such that their vector sum equals zero. Then, if a vector equals zero, the magnitude of each of its components must equal zero - this produces three scalar equations to solve. A second technique (used here) is to scalar multiply the vector equation successively by I, then J, then K . The result is the same three scalar equations. See below:

The three unknown quantities are |0P|, |BP| and H. Manipulating these equations one obtains:

The apparatus dimensions were chosen symmetric to make this symmetric and minimize calculations. Typically there are students who visually see the symmetry and proclaim proudly and correctly, "The answer is two feet." Symmetry is wonderful when its there and when you see it. The vector method must be learned. Tomorrow the dog trainer wants the wire BC located to pass wire 0A with the closest distance between them being 0.25 meters (the height of his smallest dog). Also he hates symmetry. The structure dimensions must be changed to 8.3 m x 17 m x 11 m. So, just like us, the sharp student needs patience, hard work and vectors to get there. Bon voyage!