Trajectories on circles and spheres Determine whether the...

Trajectories on circles and spheres Determine whether the following trajectories lie on either a circle in $\mathbb{R}^{2}$ or a sphere in $\mathbb{R}^{3}$ centered at the origin. If so, find the radius of the circle or sphere, and show that the position vector and the velocity vector are everywhere orthogonal. $$\mathbf{r}(t)=\langle 8 \cos 2 t, 8 \sin 2 t\rangle, \text { for } 0 \leq t \leq \pi$$

Trajectories on circles and spheres Determine whether the following trajectories lie on either a circle in $\mathbb{R}^{2}$ or a sphere in $\mathbb{R}^{3}$ centered at the origin. If so, find the radius of the circle or sphere, and show that the position vector and the velocity vector are everywhere orthogonal.
$$\mathbf{r}(t)=\langle 8 \cos 2 t, 8 \sin 2 t\rangle, \text { for } 0 \leq t \leq \pi$$

Key Concepts

Orthogonality of Position and Velocity Vectors

When a particle moves along a circle centered at the origin, the position vector is always perpendicular to the velocity vector. This orthogonality is demonstrated by showing that the dot product between the position and velocity vectors is zero, which is a key property of circular motion.

Differentiation of Vector Functions

Differentiating a vector function with respect to its parameter provides the velocity vector, which conveys the speed and direction of motion along the trajectory. This process is fundamental in analyzing the dynamics of motion in both two and three dimensions.

Parametric Equations of Curves

Parametric equations express the coordinates of points on a curve as functions of a parameter. This method captures how curves evolve over time and is especially useful in describing trajectories in the plane or space, such as circles or spheres.

Circular Trajectories in the Plane

A circular trajectory is one where the position function traces out a circle that is typically defined by an equation of the form x^2 + y^2 = r^2. Recognizing a circle from its parametric representation requires identifying that the sum of squares of the coordinates is constant, thereby determining the radius of the circle.

Transcript

00:02 Hello, in this problem we are given a position function.

00:06 R of t is a vector function given in component form, 8 cosine 2t, and the second component is 8 sign of 2t.

00:24 Are ready for t between 0 and pi.

00:31 Between 0 and pi.

00:35 Alright.

00:38 The first thing to notice is that this is in two dimensions.

00:48 So when we have a vector of position, then it starts at the origin and terminates at some point xy right so let's take this is x this is y and we suspect actually we have to test is does this point do all these points where these vectors are t point do they belong to a circle of sum of some radius so like square a cosine to and 8 sign 2t and see what we get alright so this is 64 cosine squared of 2t plus 64 of cosign squared 2t plus 64 of cosine squared pardon sign squared 2 t factor out 64 we're left with cosine square 2t plus sine square 2t now this in the parentheses this is the fundamental pythagorean identity this is equal to 1 so this expression here for any t is equal 64 which is 8 squared therefore therefore the endpoints of the position vector lie on a circle that has the equation x squared y squared equals 8 squared which is a circle of radius four lying on the origin okay so the first quarter question is to determine whether the following trajectories lie either in a circle in this is the case of two -dimensional r squared and the answer to the first question is yes the trajectory of the position vector is in fact on a circle that has radius of 8 in the plane x y plane the second thing is we need to show that the position vector and the velocity vector, now what is velocity? velocity is the derivative of the position function and the derivative of the position function is the derivative of the two components.

04:06 So let's have a side note here and find.

04:10 What is the derivative of 8 cosine 2t? alright, now this here is a composition of functions.

04:20 So the derivative is going to be the derivative of cosine is minus sine of 2t times the derivative of 2t which is 2.

04:33 Altogether it's minus 16 sine of 2 t...

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