# Introduction to Parametric Equations

Parametric equations offer a powerful mathematical framework for representing curves and surfaces in terms of independent parameters. Unlike traditional Cartesian equations, parametric equations express each coordinate as a function of a separate parameter, providing a more flexible approach to describing complex geometries. This introductory overview aims to elucidate the fundamental concepts underlying parametric equations, their applications in various fields, and their significance in modern mathematics and engineering. By delving into the principles and applications of parametric equations, readers will gain a deeper understanding of their utility and versatility in modeling dynamic phenomena and solving intricate mathematical problems.

- The surface #z=xsqrt(x+y)# intersects the plane y=3 along a curve c, how do you find the parametric equations for the tangent line to this curve at the point P(1,3,2)?
- Consider the line which passes through the point P(-1, -5, 4), and which is parallel to the line x=1+5t y=2+5t z=3+5t. How do you find the point of intersection of this new line with each of the coordinate planes?
- A curve in the xy-plane is defined by the parametric equations #x = t^3 + 2# and #y = t^2 - 5t# how do you find the slope of the line tangent to the curve at the point where x = 10?
- Given #x^2 + (y – 2)^2 = 4 # how do you derive a parametric equation?
- The curve given by the parametric equations #x=16 - t^2#, #y= t^3 - 1 t# is symmetric about the x-axis. At which x value is the tangent to this curve horizontal?
- For #f(t)= (e^t/t,4t+1/t)# what is the distance between #f(2)# and #f(5)#?
- How do you graph the curve whose parametric equations are given and show its orientation given #x = sqrt{t} + 4#, #y = sqrt{t} - 4#, where #t>=0#?
- How do you write a vector equation and a parametric equation for each line: the line through A(3,0,4) and parallel to the x-axis?
- How do you find the parametric equations for the line through the point P = (2, -2, -1) that is perpendicular to the plane 1x + 3y - 2z = 1?
- For #f(t)= (1/(t-3),t^2)# what is the distance between #f(0)# and #f(2)#?
- What is parametric equation of the line created by the intersecting planes x = 2 and z = 2?
- How do you find the angle between the planes 2x+5y-z=6 and 3x-2y+6z=10?
- How do you write the cartesian equation for x = t - 2 and y = -(t²) + t + 1?
- For #f(t)= (t^3-t+1,t^2-t)# what is the distance between #f(2)# and #f(5)#?
- For #f(t)= (t-2,-t^2-2t)# what is the distance between #f(2)# and #f(5)#?
- Given x(t)=3sin(t) - 3, y(t)=t-1 for 0 is less than or equal to t is less than or equal to 2pi How do you find the position of the particle at t=3?
- How do you find the equation of the tangent line for the curve given by x = 2t and #y = t^2 + 5# at the point where t = 1?
- How do you find the parametric equations of a circle?
- Consider the parametric equations x = 3t - 5 and y = 2t + 3 how do you eliminate the parameter to find a Cartesian equation of the curve?
- Let l be the line that passes through the point P=(−5,−7,−1) and is perpendicular to the plane 8x−6y−1z=15, how do you find the parametric equations?