Sketch the curve represented by the vector-valued function and give the orientation of the curve

Sketch the curve represented by the vector-valued function...

Labs

Want to see this concept in action?

NEW

Explore this concept interactively to see how it behaves as you change inputs.

View Labs

Key Concepts

Vector-Valued Functions

A vector-valued function assigns a vector to each value of a parameter, often representing points in a space. It is a foundational tool in calculus and physics for describing trajectories and curves by specifying coordinates as functions of a parameter.

Parametric Equations

Parametric equations describe a curve by defining each coordinate (such as x, y, and z) in terms of an independent parameter. This method allows for the representation of complex curves and motions in space, providing a clear framework to analyze and sketch their behavior.

Lines in Three-Dimensional Space

When the coordinate functions in a vector-valued function are linear functions of the parameter, the resulting curve is a straight line in space. Such a representation includes a point and a direction vector, which together fully describe the line's position and orientation in three dimensions.

Curve Orientation

The orientation of a curve indicates the direction in which the curve is traversed as the parameter increases. It is determined by the order of the parameter values and the corresponding points on the curve, and in the context of a vector-valued function, it is often reflected in the sign and magnitude of the derivative (or velocity) vector.

Transcript

00:01 We're going to sketch the curve represented by the vector valued function.

00:04 So let's make a table.

00:05 We'll have our t values and then we'll be able to find our x, y, and z.

00:11 So i do notice, though, as i get started here, and i'll just throw some t values in, look at our i, j, and k component, our x, y, and z pieces.

00:23 Notice how they are all linear.

00:26 So when all three of them are linear, the result is also going to be linear.

00:31 So as i plot my points, i am going to kind of make sure that they also are in a straight line because i do know my result should be a straight line.

00:40 Okay, so putting t into x is easy because x equals t.

00:45 So i can just rewrite all my t's.

00:49 Now, linear is nice because you can put your value in your first one.

00:52 I have a negative 2 times 2 that's negative 4 minus not.

00:56 Or minus five is nine.

00:59 And then because i have that rate of two, that is my slope, then each time i should get a value that's too higher.

01:07 So negative seven, negative five, negative three, and negative one.

01:11 However, you can put each of those values in, and you can see that that works.

01:16 And then now i'm putting it into three t, so i just multiply everybody by three.

01:21 So i'll get a negative six, a negative three, a zero, a three, and a six.

01:25 Okay...