Points, Vectors, and Functions
Recall that functions are much like computers or machines that take in one or several input numbers and put out a single number. And recall that vectors are mathematical entities composed of two pieces, magnitude and direction, like the mythological minotaur.
Now imagine trying to back the minotaur into a corner and shove him into your computer program/function machine, where he'll be torn to shreds and something else will come out. Talk about a mess.
We're used to functions whose input is one number and whose output is one number. We say such a function f maps from the real numbers to the real numbers, written
f(x) = x2
f(x) = 5
There are also functions whose input consists of multiple numbers (or a vector), while the output is still one number. In cases like this, the minotaur becomes an ordinary mouse, composed of a single, tame animal.
Let f(x,y) = x + y.
This is a function that takes two numbers as input and gives only one number as output. Evaluate f(1,5) for practice: f(1,5) = 1 + 5 = 6
Since this function takes two real numbers and gives back one real number, we say f:ℜ2→ℜ
The function f(x,y) = x + y takes two numbers as input.
This is the same thing as taking one 2-dimensional vector as input:
Remember that real numbers are also called scalars. A function whose output is a real number is a scalar function. It doesn't matter what the domain of f is. Any function f:?→ ℜ is a scalar function.
A function whose output is a vector is a vector function. Vector functions are also called vector-valued functions.
For example, a function takes in a lion and an eagle, two animals, which we equated to scalars. The function outputs a griffin, a hybrid of the two animals, and one we equated to a vector.
A vector function can also take in vectors as inputs and produce vector outputs. A griffin plus a minotaur equals a griffotaur.
Be Careful: Whether a function is a vector function or a scalar function depends on the output of the function, not the input.
Since there are different kinds of notation for vectors,
there are also different kinds of notation for vector functions.
Scalar and vector functions can be useful for thinking about real-world problems.
Here are some examples of scalar and vector functions that might be useful to someone, somewhere, maybe even outside of a calculus class.Examples of scalar functions:
Pretend a mountain is sitting on an x,y coordinate grid:
The function that takes a point (x,y) on the grid as input and gives the elevation of the mountain at point (x,y) as output is a scalar function.
This function is closely related to contour maps.
Impose a 3-dimensional coordinate grid on a room.
There's a scalar function that takes a point (x,y,z) in the room as input and outputs the temperature at that point in the room.
The function that sends a 2-dimensional vector (or an n-dimensional vector) to its magnitude is a scalar function.
The function that projects a vector onto its x-component is a scalar function:
fx(x,y) = x
As is the function that projects the vector onto its y-component:
fy(x,y) = y
Examples of vector functions:
The function whose input is a vector and whose output is a unit vector with the same direction is a vector function.
Put a coordinate grid over an etch-a-sketch.
Start a stopwatch and start drawing. At time t>0 the "pen" of the etch-a-sketch is at some point (x(t),y(t)).
The function whose input is t and whose output is the point (x(t),y(t)) is a vector function. This idea will be useful when we get to parametric equations.