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Active Calculus - Multivariable

Nicholas Long, Mitchel T. Keller, Steve Schlicker

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Preview Activity 9.2.1.
Postscript is a programming language whose primary purpose is to specify how to generate text or graphics. The following is a simple set of Postscript commands that produces the triangle in the plane with vertices \((0,0)\text{,}\) \((1,1)\text{,}\) and \((1,-1)\text{:}\) 
 
              (0,0) moveto 
              (1,1) lineto stroke
              (1,-1) lineto stroke
              (0,0) lineto stroke
The process described by these commands is
  1. tell Postscript to start at the point \((0,0)\text{,}\)
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  2. draw a line from the point \((0,0)\) to the point \((1,1)\) (this is what the line to and stroke commands do),
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  3. draw lines from \((1,1)\) to \((1,-1)\text{,}\) and
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  4. \((1,-1)\) back to the origin.
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Each of these commands encodes two important pieces of information: a direction in which to move and a distance to move. Mathematically, we can capture this information succinctly in a vector. To do so, we record the movement on the map in a pair \(\langle x, y \rangle\text{,}\) where \(x\) is the horizontal displacement and \(y\) the vertical displacement from one point to another. This pair \(\langle x, y \rangle\) is a vector, and we call each number in the pair a component. As an example, the vector from the origin to the point \((1,1)\) is represented by \(\langle 1,1 \rangle\) while the vector from the point \((1,1)\) to the origin is represented by \(\langle -1,-1 \rangle\text{.}\)
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(a)
What is the vector \(\vv_1 = \langle x , y \rangle\) that describes the displacement from the point \((1,1)\) to the point \((1,-1)\text{?}\)
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(b)
How can we use the two components of \(\vv_1\) to determine the distance from the point \((1,1)\) to the point \((1,-1)\text{?}\)
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(c)
Suppose we want to draw the triangle with vertices \(A=(2,3)\text{,}\) \(B=(-3,1)\text{,}\) and \(C=(4,-2)\text{.}\) As a shorthand notation, we will denote the vector from the point \(A\) to the point \(B\) as \(\overrightarrow{AB}\text{.}\) Determine the vectors \(\overrightarrow{AB}\text{,}\) \(\overrightarrow{BC}\text{,}\) and \(\overrightarrow{AC}\text{.}\)
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(d)
How are the horizontal displacements of \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\) related to the horizontal displacement of \(\overrightarrow{AC}\text{?}\) Does the same relationship follow for the vertical displacements? Write a couple of sentences to explain your reasoning in the context of this problem.
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