 A function takes one or more inputs and produces an output. You've worked single variable functions that produce a single numerical output, the height of a ball at time t. You also have looked at multivariable functions that produce a single numerical output, although you might not have recognized them as multivariable functions. For example, area is a function of length and width. But who says the output has to be a number? So we introduce the idea of a vector-valued function is a function whose output is a, wait for it, vector. For example, f of t is the vector t times 3, 4, 1, and let's find f of negative 1. Now you've worked with function notation before and this isn't really any different. We replace and evaluate. So we have our function. We'll drop our input variable and replace it with an empty set of parentheses and whatever we put in one set of parentheses goes in all of them. And we compute. And the only thing that's really different here is that the output is a vector and not a single numerical value. So it's helpful to remember our geometric interpretation of a vector. A vector gives us the directions we're getting from a point to another point. The starting point is the origin. The components of the vector are the same as the coordinates of the point and this allows us to write equations in vector form. For example, let's try to write a vector-valued function for the line through 2, 5, and 1, 8. So our output should be a vector that goes from the origin to a point on the line. Since 2, 5 are the coordinates of a point, we can use the vector 2, 5 to go from the origin to a point on the line. And remember a vector gives you the directions for going from a point to another point. So we can go from one point on the line 2, 5 to another point on the line 1, 8 by following the vector, negative 1, 3. And we can get to other points by going any scalar multiple of this vector. And so we get our vector equation, the vector 2, 5, plus a scalar multiple of the vector 1, negative 3. Let's try to find three different vector equations for the line through 1, 3, negative 1, 6, and 2, 8, 4, negative 5. So we should try and graph the points, except they're in four dimensions and we don't have four-dimensional graph paper. We can still write the vector equation as long as we remember a vector gives you the directions for going from a point to another point. So remember if our starting point is the origin, the vector components and the coordinates of a point will be the same. And so if I want to get to this point 1, 3, negative 1, 6, the vector 1, 3, negative 1, 6 will take us from the origin to a point on the line. Now we know another point on the line is 2, 8, 4, negative 5. And so we can go to another point on the line by traveling along the vector from 1, 3, negative 1, 6, 2, 2, 8, 4, negative 5. And this vector will be, and so traveling any multiple of this vector will take us to some other point on the line. So a vector equation will be 1, 3, negative 1, 6, go to a point on the line and then go some scalar multiple of the direction of the line. How can we get a different vector equation of the line? Well, we could go to a different point. So this time we could have gone to the point 2, 8, 4, negative 5 instead. And so the vector 2, 8, 4, negative 5 will take us to a point on the line. And then we need to travel along the vector from 2, 8, 4, negative 5 to 1, 3, negative 1, 6, which will be, and again if we go any scalar multiple of that vector we'll go to a different point on the line. How about a third equation? Well, we always start out by going to some point on the line and we have two points on the line. We could have gone to a point determined by either equation for some value of t. So for example, if t equals 3, our first equation will give us... So remember that this vector takes us from the origin to a point on the line and so our point will have the same coordinate as the vector. So we'll be at 4, 18, 14, negative 27. And we need a vector that takes us from a point on the line to another point on the line. And that's the whole purpose of this part of the vector equation. And so we note that the vector 3, 15, 15, negative 33 takes us from a point on the line to another point on the line and so our vector equation will go any scalar multiple of this vector. Now you might be a little concerned notice that our three vector equations were completely different. Does it really matter? And it turns out the three vector equations will all go through the same points so they describe the same line. Now you can prove this algebraically. And you should. But you might also remember that f and g at least go through the same two points and so they should be the same line. You also know that this point 4, 18, 14, negative 27 is actually on the line defined by f. And if you think about it, that means that f and h also go through the same two points and so they must also be the same line.