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. $$\mathbf{r}(t)=\langle 3 \sin t, 5 \cos t, 4 \sin t\rangle, \text { for } 0 \leq t \leq 2 \pi$$

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.
$$\mathbf{r}(t)=\langle 3 \sin t, 5 \cos t, 4 \sin t\rangle, \text { for } 0 \leq t \leq 2 \pi$$

Key Concepts

Dot Product and Orthogonality

The dot product of two vectors provides a measure of the angle between them. When the dot product equals zero, the vectors are orthogonal (perpendicular). In the context of trajectories, showing that the position vector and the velocity vector are orthogonal means that the rate of change of the distance from the origin is zero, which further confirms the constancy of the radius and the circular or spherical nature of the motion.

Radius Determination

To determine the radius of a circle or sphere, one computes the magnitude (or norm) of the position vector derived from the parametric equations. If this magnitude remains constant for all parameter values, the trajectory lies on a circle or sphere with the constant magnitude serving as the radius, which is fundamental to understanding the geometric structure of the path.

Parametric Equations

Parametric equations represent curves or surfaces by expressing the coordinates of the points as functions of one or more parameters. They allow the description of trajectories, such as circles or spheres, through functions like sine and cosine, which inherently capture periodic motion and symmetry around a central point.

Circles and Spheres Centered at the Origin

A circle in two-dimensional space and a sphere in three-dimensional space centered at the origin are defined as the set of all points that maintain a constant distance, called the radius, from the origin. This constant-distance property is key when determining if a given trajectory lies on a circle or sphere, as it requires verifying that the norm of the position vector is invariant over time.

Transcript

00:01 Hello, this problem gives us a position function, a position function r of vector value r of t is three components so we are in the three dimensional space.

00:21 First component is 3 sine t the second component is 5 cosine t and the third is 4.

00:33 Sign t for t between zero and two one we need to find does the trajectory lie on a circle a circle centered at the origin has an equation like this so we will be squaring anyway second if so we need to find the radius and prove or test that the velocity vector of t is orthogonal to the position vector of t so in three -dimensional space let's say that this is one of the vectors r of t, r of t, the beginning point is at the origin and the end point of each of these is in a point that has coordinates xyz x, y, z.

01:53 To check whether these points lie on a circle of a certain radius, this would have to be constant.

02:01 Right so let's square x squared plus y squared plus z squared equals the square of this is 9 sine square t plus the square of this is 25 cosine square t and the square of this is 16 sine square t like terms there are two of them here so we have 25 sine square t plus these 25 cosine square t factor out 25 and we're left with the fundamental pythagorean identity of trigonometry this is one so this is 25 this is 5 squared so the answer to the first question is yes a circle is for the two -dimensional or for the three -dimensional i'm sorry i should have said this is the equation of a sphere in three dimensions a circle would be this equation with that is it so yes the trajectory lies a sphere centered on a sphere centered on a sphere with radius 5 okay onwards to find to check whether the vectors of the vectors of velocity and position are orthogonal first to find the vector of velocity of t is the derivative of the position of t which is we go back to each of these components and find the derivatives of each of them so the the derivative of sine is cosine, so the derivative of this would be 3 cosine t.

05:04 This would be the derivative of cosine is minus sine, so it would be minus 5 sine t...

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