Match the vector equations with the graphs (a)-(h) given here. (GRAPH CANT COPY) 𝐫(t)=𝐢+𝐣+t 𝐤,

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

Key Concepts

Line Segments versus Infinite Lines

In vector equations, the parameter's domain determines whether the equation represents a complete line or a restricted segment. With t bounded between -1 and 1, the equation defines a finite line segment rather than an infinite line, highlighting the importance of parameter restrictions in describing curve lengths in space.

Parameter Domain Constraints

The range of the parameter (in this case, t ? [-1,1]) limits the set of points that are included on the graph. This constraint is fundamental when identifying the correct graphical representation, as it restricts the vector equation to only a portion of the full line that would otherwise extend indefinitely if the parameter were unbounded.

Parametric Vector Equations

This concept involves expressing curves in space by representing the coordinates (x, y, z) as functions of a parameter, typically t. The equation r(t) = i + j + t k is an example where the constant components provide a fixed starting point, and the term with the parameter represents the direction and movement along the curve.

Graphical Interpretation in Three Dimensions

Understanding how to translate a vector equation into a visual graph is crucial. The components of the equation correspond to spatial coordinates, enabling one to interpret the position and movement within a three-dimensional coordinate system. This is essential for matching equations to their geometric representations.

Transcript

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

00:10 So let me just copy down this vector equation here.

00:13 We have this vector equation that says r as a vector value function gives you i, excuse me, i'm going to write i, j and k as x had, y, had, and z had.

00:28 That's something that i can do.

00:29 So i'm going to write x had plus y had plus t times z had these three are just unit vectors x had y had and z had and our time interval is restricted to from negative one from negative one to one to one okay so now let's take a look at this vector value function let's take a look at its components and see what's going on here.

01:00 So first of all, i want you to look at this this x component and y component right here.

01:05 The coefficient in front of the two unit vectors are actually one.

01:10 They're constant.

01:11 They're not changing with time.

01:13 So what that means is that during your whole time interval, your x and y component, they shouldn't change.

01:20 They should stay one.

01:22 So the x, for all values of t between negative 1 and 1 should be 1, and y for all values of t should be one and z as you can see is a linear function it grows linearly what that means is that if we were to graph this function out this is the x -axis this is the y -axis and this is the x -axis and i say that this is the this is the value one here and the value one here if you were to graph x equals y plus one that point is going to be right here and we want the value of z to grow linearly.

02:05 When z is equal to negative 1, the z component of the function r is going to be negative 1.

02:13 And so we're going to start off right here...

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