Match the vector equations with the graphs (a)-(h) given here. (a To h Figure can't copy) 𝐫(t)=t

Match the vector equations with the graphs (a)-(h) given...

Key Concepts

Endpoint Interpolation

Endpoint interpolation involves calculating the positions corresponding to the extreme values of the parameter in the equation. This identifies the endpoints of the line segment, demonstrating how the vector equation transitions from one endpoint to the other as the parameter varies over its interval.

Line Segments

When the parameter in a vector equation is restricted to a closed interval, the resulting graph is typically a line segment. The endpoints of the segment are determined by substituting the endpoint values of the parameter interval into the equation, yielding the starting and ending points of the segment.

Parametric Representation

Parametric representation is the technique of describing a curve (or surface) by introducing one or more parameters. Each coordinate is expressed as a function of these parameters, enabling the representation of complex shapes and the tracking of a point moving along the path.

Vector Equations

Vector equations represent geometrical objects such as lines or curves by expressing the coordinates of points as functions of a parameter. This form allows for a concise description of the position of points in space and is fundamental to understanding curves and surfaces in both two and three dimensions.

Transcript

00:01 Okay, folks, so in this video, we're going to talk about this problem where we're given a vector equation and we want to match it with one of the graphs that are given here.

00:12 And the graphs go from a to h.

00:15 Okay, so we are going to write down this vector equation.

00:18 So first of all, we have r as a function of t.

00:23 This is a vector function, vector value function, parameterized by the variable t.

00:28 And the x component of this function is given by t and the y component of this um this function is given by one minus t and i just want you to know that the z component is not mentioned and when when the problem doesn't mention the z component that means it's zero so the z component of this um of this vector value function is zero um at least for the time interval of um zero less than or equal to t than are equal to 1.

01:01 Okay.

01:01 So in this time interval, there is no z component.

01:04 What that means is that all other things aside, at least we know, one thing that we definitely know is that this graph of this function is definitely going to lie in the x, y, axis.

01:18 Okay, so that's what we know for sure.

01:21 So just by that, we can eliminate a lot of the choices.

01:26 So a lot of the graphs where the z component is not 0 is going to be eliminated.

01:32 But now let's try and relate the x and y component of this of this vector function.

01:38 So we know that the y component or the j component is 1 minus t.

01:43 So i'm going to write y equals 1 minus t and we know that the x component of this function is is t.

01:54 So there is as you can see there's a little bit of a relationship between y and x and that relationship.

01:59 And that relationship can be written as y equals 1 minus t but t is x so we have this this is the relationship between y and x of this vector valued function so we can graph it out i mean i i definitely hope that you know how to graph this simple you know line out we have x x x is here and y axis here then y equals 1 minus x is just going to give you a line whose y intercept is 1 and and x intercept is also one...

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