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

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

Key Concepts

Constant Magnitude and Radius Determination

Establishing that the magnitude of the position vector remains constant is vital to showing that a trajectory lies on a circle or sphere. The process involves calculating the distance from the origin to the point described by the parametric equations and verifying that it does not vary with the parameter. This constant value directly indicates the radius of the circle or sphere.

Dot Product and Orthogonality of Vectors

The dot product is an algebraic operation that, when applied to two vectors, yields a scalar. It is especially useful for determining whether vectors are perpendicular; if the dot product of two non-zero vectors is equal to zero, then the vectors are orthogonal. In the context of motion, demonstrating that the position and velocity vectors are orthogonal implies that the motion is tangential to the circular path.

Circles and Spheres in Euclidean Space

Circles (in ?²) and spheres (in ?³) are sets of points that maintain a constant distance, known as the radius, from a fixed center point (often taken as the origin). Recognizing when a parametric equation describes such a set involves confirming that the magnitude of the position vector remains constant over the parameter range.

Parametric Curves

Parametric curves involve defining the coordinates of points on a curve as functions of a parameter (typically time). This concept allows us to describe complex trajectories in the plane or space and analyze their properties, such as curvature and orientation, by examining how the coordinates change with respect to the parameter.

Differentiation of Vector-Valued Functions

Differentiation of vector-valued functions is used to compute the velocity vector by taking the derivative of the position vector with respect to the parameter. This concept is critical to understanding the instantaneous direction and speed at which the point is moving along the trajectory.

Transcript

00:01 Hello.

00:03 We are given a position function.

00:07 This is in the two -dimensional case of r2.

00:11 The position function is a vector function of t in component form, four sign t, and the same component is two cosine t.

00:28 2 cosine t on an interval for the 0 to 2 pi 2 questions 1 is it does it lie on a circle if it does the radius and secondly we have to show that the velocity vector is orthogonal to the position vector which is defined with dot product being zero.

01:13 Okay, in an xy plane, if we have the vector rt, the terminal point of each vector is some point xy, so we take x and y to be these two, and the trajectory is going to be wherever these vectors are pointing four values of t from 0 to 5.

01:44 Okay, circle centered at the radius has an equation x squared plus y squared equals r squared and our first job is to find out whether the points xy satisfy this thing here.

02:03 So squaring x, squaring y, we obtain 16 sine squared t plus 4 cosine square t and we can write the 16 sine squared as 12 sine square t plus 4 sine square t and we add the four cosine squared here in order to extract four a factor out four here this will be sine squared t plus cosine squared of t 12 sine squared pythagorean no actually this was a plus and this was a t sine square of t plus cosine square of t is the fundamental pythagorean equation identity this equals 1 so x squared plus y squared is not constant it changes depending on t.

03:42 So it's 12 sine squared t plus 4 which is not equal to some radius.

03:55 So this here is what is this is probably an ellipse but it's not a circle...

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