Match the vector equations in Exercises 1-8 with the graphs (a)-(h) given here. 𝐫(t)=(2 cost) 𝐢+

Match the vector equations in Exercises 1-8 with the graphs...

Key Concepts

Circle and Semicircle Representations

The equation r(t) = (2 cos t)i + (2 sin t)k describes a circle of radius 2 in the plane determined by the i and k unit vectors. Limiting the parameter t to [0, ?] results in a semicircular arc, which is key for correctly identifying the associated graph.

Vector Functions

A vector function assigns a vector to each value of a parameter, representing curves or trajectories in space. This concept is central to understanding how an equation like r(t) = (2 cos t)i + (2 sin t)k traces out a path as t varies.

Parametric Equations

Parametric equations express the coordinates of points on a curve as functions of a parameter, rather than directly relating x, y, and z. This framework is crucial for representing curves that cannot be described as functions in the conventional sense.

Trigonometric Parametrization

This method uses sine and cosine functions to define the coordinates of points on curves, particularly circles and ellipses. It captures the periodic behavior inherent in such shapes and is a common strategy for depicting circular paths.

Graphing Parametric Curves in Space

Graphing involves plotting the points generated by a parametric equation in a coordinate system to visualize the curve. Understanding how these graphs are generated aids in matching equations with their visual representations.

Transcript

00:03 Here we are asked to match a vector equation with one of eight graphs.

00:07 Let's notice a few things.

00:09 First, our vector equation is given in parametric form.

00:13 The variable t plays an indirect role in establishing a relationship between the vector variables i and j.

00:22 Also, when we look at the possible graphs, we see that they're in three dimensions.

00:27 We've got an x -axis, a y -axis, and a z -axis.

00:30 So even though we don't see a variable k in the vector equation, we need to imagine it's there.

00:41 So i will add a term for the vector k, 0 times k, to remind us that we're in three dimensions.

00:56 Now let's write the vector equation as scalar equations.

01:02 We'll have three of them, one for x, which is the component of the i.

01:08 Vector y which is the component of the j vector and z will always be zero so let's take some points and put them on a graph and see what we come up with so we have our t parameter and our x y and z variables notice that our variable t parameter t is defined between zero to pi so we can take some sample where t is equal to 0, pi over 2, pi, 3 pi over 2, and then 2 pi.

02:19 When t equals 0, we've got cosine of t equal to 1, multiplied by 2 gives us x equals 2.

02:33 For y, we have sine t which is equal to 0, 2 times 0 is 0.

02:43 Z is always equal to 0.

02:45 So we'll dispense with that.

02:50 If pi over 2, cosine of t is equal to 0, so x will be 0, sine of pi over 2 is 1, 2 times 1 is 2.

03:04 At t equals pi, we have cosine equal to minus 1, giving us minus 2...

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