# vectors:

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coordinate space, vector basis and operations.

## 1.02 Position: the First Vector

"Where is something" is a question persons have asked over all of time. The answer tells the "position of that something" in a mutually understood space. The "something" might be your car keys or a location such as a freeway exit or the rest room, etc. For a person to answer, "Where is it?," requires both familiarity of both persons with the surrounding space. The space "it is in" must be mutually known. ## Pharaoh's Engineers

The Great Pyramid of Egypt was constructed to precise proportions. A hypothesis is that the pyramid was constructed to fit inside an imaginary hemisphere with each of its corners and its peak touching the hemisphere. Suppose the hypothesis were true. Draw a sketch of a pyramid and calculate the angle each face makes with the horizontal plane of the desert.

## Vectors Derive Trig Laws A typical planar triangle is shown. The triangle vertices, lengths of sides, and angles are identified by notation.

We choose to define a vector space to be the plane of the triangle. We name the two-dimensional space 0XY. We place the origin of the space at the A0B vertex of the triangle and we align its X-axis along the triangle side, 0A... ## Crank, Rod and Piston

"Power trains" are the mechanical parts by which the power of combustion is transformed into power of a rotating shaft. A sketch of a simple crank-rod-piston arrangement is shown. Engine designers must know the position of the piston face for every position of the crank. The mathematical tool, vectors, makes this task logical. Imagine a retirement home is ablaze and flames have backed two elderly women to the opening of a rear, third story window. A ladder truck has arrived in a back alley but a building obstructs its reach.

To place the ladder at the window, it will be extended nearly vertical to the correct, laser determined, length L, then tipped downward to the angle θ. Lasers will obtain the dimensions (given in the sketch) and a computer will solve the geometry. To provide a test case, use the information of the sketch.

## Quick Return The mechanism O-A-B has two links and ground. The input, Bar-OA, oscillates through small angles,­ α ± Δα. Bar-AB, slot-connected to Bar-OA at A, has a changing, active length, X(t). Figure 1. is the basic configuration with notation. Write a general "specification" for all positions of this planar mechanism. Use a math/physics approach that leads to a solution statement readily expressible in computer code.

## High Wire Apparatus After performing with ponies, the dogs of a traveling show do feats on wire strung on a cubical steel frame. The frame measures ten meters on each edge. Once it is erected cables are strung for specific dogs or tricks. Since the feats of the dogs must move smoothly and promptly, the rigger must have a good knowledge of cable lengths and places of connection. In this example we apply vectors to that task.