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

Straight Line in 3D

The equation r(t) = t i + t j + t k defines a straight line in three-dimensional space. The coefficients of t in each component indicate that the line has a constant direction, specifically along the vector (1, 1, 1), and it passes through the origin, which is a key characteristic of a line in vector geometry.

Domain Restriction of the Parameter

Imposing the condition 0 ? t ? 2 on the parameter t restricts the set of points to a specific segment of the straight line. This domain limitation specifies the endpoints of the graph, effectively defining a line segment rather than the entire infinite line.

Vector Equation

A vector equation expresses the position of points in space as a function of a parameter (often time). In the provided equation, each component of the position vector is scaled by the parameter, illustrating how the point moves through space as the parameter t varies.

Parametric Representation

The given equation is written in parametric form, meaning that the x, y, and z components are each written in terms of the parameter t. This allows us to describe the path of the point in three-dimensional space, and serves as a fundamental technique for representing curves and lines in vector calculus.

Transcript

00:01 Okay, folks, so in this video, we're going to have this vector equation that we want to match with one of the graphs from a to h.

00:09 Okay, so let me write down this vector equation for you.

00:12 We have r as a function of t, which is a vector value function, and it's going to give you t in the x component plus t, y component, plus t in the z component.

00:26 And we're going to be restricted with the following restriction, which is zero less than or equal to t, less than or equal to two.

00:39 So there's a couple of things to notice here.

00:42 The first thing is the fact that we have three linear functions.

00:50 There's no t squared or t -qued or whatever.

00:52 We have three simple -looking, nice linear functions.

00:57 So, and, and what's the most important with this is that the x component and y component and the z component of this vector value function is the same.

01:09 So we have x equals y equals z.

01:13 So all of our points that's going to be, that's going to be drawn on the graph, has equal values for the x and y and z components.

01:23 So basically what i'm saying is it's supposed to be a line.

01:26 And it's aligned with with some very nice properties for example the three components are are the same and it starts off at at the origin because when t is zero all of the components are zero and so we can graph it out we can go ahead and see if we can grab it out so this is x and this is y and this is z when x when t is zero we are located at the origin so that's the origin that's when t is 0.

01:56 And when t is 2, we have x equals y equals z equals 2.

02:01 So i'm going to graph that out.

02:03 We have 2 here and 2 here and 2 right here...

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