Preview Activity 13.1.1.
It’s common when discussing weather to talk about the wind speed, but as any student who has gotten this far in the text will know, this nomenclature is imprecise. It’s not terribly helpful to tell someone the wind is blowing at 10 km⁄h without telling them the direction in which the wind is blowing. If you’re trying to make a decision based on what the wind is doing, you need to know about the direction as well. For instance, if you are taking off in a hot air balloon, the wind direction will determine which direction the chase team should start going to keep track of you. Because of the swirling nature of wind, it makes sense to give the wind velocity at each point in a region (two-dimensional or three-dimensional).
(a)
Suppose that given a point \((x,y)\) in the plane, you know that the wind velocity at that point is given by the vector \(\langle y,x\rangle\text{.}\) For example, we’d then know that at the point \((1,-1)\text{,}\) the wind velocity is \(\langle -1,1\rangle\text{.}\) We will give the wind velocity as a function \(\vF\text{,}\) where \(\vF(x,y) = \langle y,x\rangle\text{.}\) In the table below, fill in the wind velocity vectors for the given points.
| \((x,y)\) | \((2,1)\) | \((0,0)\) | \((-1,2)\) | \((3,-1)\) | \((-2,-1)\) |
| \(\vF(x,y)\) |
(b)
Suppose that we associate the vector \(\vG(x,y) = -x\vj\) to a point \((x,y)\) in the plane. Complete the table below by giving the vector associated to each of the given points.
| \((x,y)\) | \((-2,0)\) | \((-1,2)\) | \((0,-2)\) | \((2,3)\) | \((3,2)\) | \((-1,0)\) | \((1,3)\) |
| \(\vG(x,y)\) |
(c)
A table of values of these vector-valued functions is useful to understand the input vs. output nature of a vector field as a function, but perhaps even better is a method of visualizing the vector outputs. A good picture is worth a thousand words (or numbers). Returning to our analogy of the output vector for our vector field being wind velocity, if \(\vF(2,1) = \langle 1,2\rangle\text{,}\) this means that at the location \((2,1)\) the wind is moving in the direction given by \(\langle 1,2\rangle\text{.}\) Thus, we draw the output vector \(\langle 1,2\rangle\) with its initial point at \((2,1)\text{.}\)
Using the first set of axes in Figure 13.1.1, plot the vectors \(\vF(x,y)\) for the five points in the table in part a. The example \(\vF(1,-1) = \langle -1,1\rangle\) is drawn for you.
(d)
Using the second set of axes in Figure 13.1.1, plot the vectors \(\vG(x,y)\) for the eight points in the table in part b.
Axes for a rectangular coordinate system. The horizontal axis is labeled \(x\) and the vertical axis is labeled \(y\text{.}\) Both axes range from \(-4\) to \(4\text{.}\) There is a blue vector pointing from the point \((1,-1)\) to the point \((0,0)\text{.}\)
Axes for a rectangular coordinate system. The horizontal axis is labeled \(x\) and the vertical axis is labeled \(y\text{.}\) Both axes range from \(-4\) to \(4\text{.}\)
