Trajectories on circles and spheres Determine whether the following trajectories lie on either a

Trajectories on circles and spheres Determine whether the...

Key Concepts

Vector-Valued Functions

A vector-valued function is a function that outputs a vector for each input value, typically representing the position of a moving particle in space. These functions are used to describe trajectories and motions in various dimensions, with their components often expressed in terms of trigonometric functions to capture periodic behavior.

Circles in Euclidean Space

A circle in the plane R² is defined as the set of all points that are at a constant distance (the radius) from a fixed point (the center). Determining if a trajectory lies on a circle involves verifying that the magnitude (or norm) of the position vector remains constant over time.

Spheres in Euclidean Space

Analogous to circles in R², a sphere in R³ is the collection of all points that are equidistant from a central point. Verification that a trajectory lies on a sphere involves showing that the position vector's norm is constant, ensuring all points are the same distance from the origin.

Norm Calculation

Calculating the norm (or magnitude) of a vector is essential in determining the radius of a circle or sphere. A constant norm across all values of the parameter confirms that the trajectory is confined to a circle or sphere with that fixed radius.

Dot Product and Orthogonality

The dot product of two vectors provides a measure of their directional alignment. If the dot product is zero, the vectors are orthogonal. In kinematics, showing that the velocity vector is always orthogonal to the position vector is a characteristic of circular motion, as it implies that the motion is always tangential to the circle.

Trigonometric Identities and Linear Combinations

Linear combinations of trigonometric functions can often be simplified using trigonometric identities. This simplification is key to revealing underlying geometric patterns, such as the constant norm of a position vector, which in turn indicates circular or spherical trajectories.

Transcript

00:02 Hello, this problem gives us a position function r of t, vector function r of t, is the first component is sine t, plus square of three, cosine t.

00:20 And the second component is square of three sine t and minus cosine t on.

00:38 4t in the interval from 0 to 2pup.

00:47 Okay, the problem wants to find does the trajectory lie on a circle? if so, then we need to find what is the radius and the third thing is to check that the velocity vector is orthogonal to.

01:11 The position vector for all t meaning that the dot product of v times r and r here doesn't it v times r is equal to zero this would mean that it's orthogonal all right so this is two components so we're talking about the r squared dimensional space where if say this is one r of t positional vector it terminates in a point xy so the coordinates of the x and the y depending on the x and y are functions of t are these now what we need to check does the trajectory lie in a circle meaning that is x squared plus y squared equal to some constant squared this would be the radius right so let's square x let's square y and see where we go square of this is we have first plus second so we're going to have on squaring a binomial this scheme here first term squared double the product of the first and second term and the second term squared so we have sine square of t plus two times a square of three sine t cost t plus the square of this is three cosine square t plus this is x the same scheme goes for here except that it's going to be minus double the product so the first term squared is three sine t minus two three sign t cost t i'll explain what we're going to do here so let's just raise this from a workspace and this was the first squared this was the product of double product of terms and the last term squared is cosine squared t let's find like terms so x squared plus y squared this is equal to that uh like 3 plus cosine squared is 1 right now let's see what we can do with these.

04:36 This is 2 times square of 3 and the square of the first term squared was sine square t.

04:56 These two terms cancel.

04:59 Sign t plus cosine t, sign squared t plus is one and the last bits that we observe are these two where we can factor out three and we have left cosine square t plus sine square t this is one so the sum of the squares of x and y is 4 which is 2 so the answer to the first question posed in the problem is yes the trajectory lies on a circle around the origin, and the radius of the circle is 2.

06:10 Okay, now we go to the second part in which we check whether v of t is orthogonal to r of t at all values of t.

06:23 To find v of t, v of t is the derivative of the velocity function, is the derivative of the position function.

06:38 And to find these, let me just, we need to find the components, the derivatives of the components of this year and this year.

06:55 Right so on the side let's see sign t plus we're doing this one first sign t plus square of three derived by by t you can write this as well same thing all right equals the the derivative of x cosine is x.

07:25 The derivative cosine is minus sine t.

07:29 So we have cosine t minus square of three sine t.

07:39 That is the first component of the velocity function.

07:53 Alright, move that further down.

08:02 And the second component is this one here.

08:05 Here, all righty...

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