段階的に解決し、人工知能を使用せず、優れた仕事を行いますご支援ありがとうございましたSOLVE STEP BY STEP IN DIGITAL FORMATDONT USE CHATGPT16. Let r(t) be the position vector in R³ for a particle that moves with constant speed c> 0in a circle of radius a > 0 in the xy-plane. Show that a(t) points in the opposite directionas r(t) for all t. (Hint: Use Example 1.37 to show that r(t) 1 v(t) and a(t) I v(t), and hencea(t) || r(t).)

Given is the position vector in for a particle that moves with constant speed in a circle of…

Answered: 16. Let r(t) be the position vector in…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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4. for the vector function 7(t) = t³i + t?j %3D a) Find the velocity, acceleration and unit tangent vector for t = 1 b) Find parametric equations for the line tangent to the graph ofŕ(t) when t = 1 c) Graph the vectors 7(1), 3(1) and å(1) on the same coordinate axis.
Suppose a, b, and c are positive real numbers satisfying a2 = b2 + c2.r(t) = acos(t),bsin(t),csin(t) , 0 ≤ t ≤ 2πThen the vector-valued functiondescribes a tilted circle (a circle in a plane that is not parallel to one of the coordinateplanes). The center of the circle is O(0, 0, 0).(a) Show that |r(t)| is constant and determine the constant. This is the radius of thecircle.(b) Find an equation for the plane that contains the circle by doing the following:(i) Find three points on the circle.(ii) Use the three points to find a normal vector to the plane containing the three points.(iii) Find an equation for the plane. Check: does the plane contain the center of the circle?(iv) Simplify the equation as much as possible. (Since a, b and c are positive real numbers, you can divide by them without dividing by zero.)
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2. Suppose a particle has a velocity is v(t) = (1,1, ge¯t) for constant g. Complete the following given the fact that the particle has known position (1) = (0,0,–3g/e). A) Use calculus to find the position vector of the particle at time t. B) Find a time when the particle intersects the xy plane. C) Determine the speed of the particle when it intersects the xy plane.
3. Let i, j, k denote the unit vectors along the three coordinate axes. Let v(t) = ti+ sintj + costk and w(t) = 3ti + 2k. (a) Compute (v.w)' (t) directly and check your answer by using the product rule (for the dot product, stated in the class). Note that the dot above, is the dot product of vectors and denotes the derivative. (b) Compute (v x w)'(t) directly and check your answer by using the product rule (for the cross product, again stated in the class). Note that the x above, is the cross product of vectors.

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16. Let r(t) be the position vector in R³ for a particle that moves with constant speed c>0 in a circle of radius a > 0 in the xy-plane. Show that a(t) points in the opposite direction as r(t) for all t. (Hint: Use Example 1.37 to show that r(t) I v(t) and a(t) I v(t), and hence a(t) || r(t).)