What is Parametric equations: Definition and 208 Discussions
In mathematics, a parametric equation defines a group of quantities as functions of one or more independent variables called parameters. Parametric equations are commonly used to express the coordinates of the points that make up a geometric object such as a curve or surface, in which case the equations are collectively called a parametric representation or parameterization (alternatively spelled as parametrisation) of the object.For example, the equations
x
=
cos
t
y
=
sin
t
{\displaystyle {\begin{aligned}x&=\cos t\\y&=\sin t\end{aligned}}}
form a parametric representation of the unit circle, where t is the parameter: A point (x, y) is on the unit circle if and only if there is a value of t such that these two equations generate that point. Sometimes the parametric equations for the individual scalar output variables are combined into a single parametric equation in vectors:
(
x
,
y
)
=
(
cos
t
,
sin
t
)
.
{\displaystyle (x,y)=(\cos t,\sin t).}
Parametric representations are generally nonunique (see the "Examples in two dimensions" section below), so the same quantities may be expressed by a number of different parameterizations.In addition to curves and surfaces, parametric equations can describe manifolds and algebraic varieties of higher dimension, with the number of parameters being equal to the dimension of the manifold or variety, and the number of equations being equal to the dimension of the space in which the manifold or variety is considered (for curves the dimension is one and one parameter is used, for surfaces dimension two and two parameters, etc.).
Parametric equations are commonly used in kinematics, where the trajectory of an object is represented by equations depending on time as the parameter. Because of this application, a single parameter is often labeled t; however, parameters can represent other physical quantities (such as geometric variables) or can be selected arbitrarily for convenience. Parameterizations are non-unique; more than one set of parametric equations can specify the same curve.
My take;
Part (a);
##\dfrac{dy}{dx}=\dfrac{1}{t}##
therefore,
##y-2at=\dfrac{1}{t}(x-at^2)##
##ty-2at^2=x-at^2##
##ty=x+at^2## implying that ##T## has co-ordinates ##(-at^2,0)##.
##SP=\sqrt{(a-at^2)^2+(0-2at)^2}##
##SP=\sqrt{4a^2t^2-2a^2t^2+a^2t^4+a^2}##
##SP=\sqrt{a^2t^4+2a^2t^2+a^2}##...
My take;
##\dfrac{dy}{dx}=\dfrac{-1}{t^2}⋅\dfrac{1}{2t}=\dfrac{-1}{2t^3}##
The equation of the tangent line AB is given by;
##y-\dfrac{1}{t}=\dfrac{-1}{2t^3}(x-t^2)##
##ty=\dfrac{-1}{2t^2}(x-t^2)+1##
At point A, ##(x,y)=(3t^2,0)##
At point B, ##(x,y)=(0,1.5t)##...
hmmmmm nice one...boggled me a bit; was trying to figure out which trig identity and then alas it clicked :wink:
My take;
##x=(\cos t)^3 ## and ##y=(\sin t)^3##
##\sqrt[3] x=\cos t## and ##\sqrt[3] y=\sin t##
we know that
##\cos^2 t + \sin^2t=1##
therefore we shall have...
Hi,
I'm trying to find the area of this tube using ##\int \int ||\vec{N}|| ds d\theta##. However, I get 0 as result which is wrong.
So at this point, I'm wondering if I made a mistake during the parametrization of the tube. This is how I parametrized the tube.
##S(s, \theta) = (cos(s), sin(s)...
Let P (1, 2, 3), Q (2, 3, 1), and R (3, 1, 2).
(a) Derive the parametric equations for the line that passes through P and Q without resorting
to the known formula.
(b) Derive the equation of the plane that passes through the points P, Q, and R without
resorting to the known formula.
(c) Find the...
Solution:
The point of tangency of the string moves around the circle at ##2\pi## radians per second. First, we compute the position of the point of tangency of the string with the bobbin. Because this is simply a revolution around a circle of radius 10, the parameterization of the point of...
11.1 Parametric equations and a parameter interval for the motion of a particle in the xy-plane given. Identify the paritcals path by finding a Cartestian equation for it $x=2\cos t, \quad 2 \sin t, \quad \pi\le t \le 2\pi$
(a) Identify the particles path by finding a Cartesian Equation the...
Homework Statement
Find the scalar, vector, and parametric equations of a plane that has a normal vector n=(3,-4,6) and passes through point P(9,2,-5)
Homework EquationsThe Attempt at a Solution
Finding the scalar equation:
Ax+By+Cz+D=0
3x-4y+6z+D=0
3(9)-4(2)+6(-5)+D=0
-11+D=0
D=11...
I am taking a high school multivariable calculus class and we have an end-of-semester project where we trace out some letters etc., except that they all have to be connected, continuous and differentiable everywhere. My group's chosen to do Euler's formula, but so far we are having problems...
Homework Statement
[/B]
Write vector and parametric equations for the line that goes through the points P(–3, 5, 2) and Q(2, 7, 1).
Homework EquationsThe Attempt at a Solution
First I find the direction vector for PQ.
PQ=Q-P = (2,7,1)-(-3,5,2)
=[2-(-3),7-5,1-2]
=5,2,-1
PQ= (5,2,-1)
Now I...
Let's say you have a helix defined parametrically as
r(t) =<sin(t), cos(t), t>
Is it possible to eliminate t and write an equation for this helix just in terms of x, y, and z?
Suppose a baseball is hit 3 feet above the ground, and that it leaves the bat at a speed of 100 miles an hour at an angle of 20° from the horizontal.
I've got the parametric equations in terms of x and in terms of y, and I have values plotted and a graph sketched. My question is in regards to...
Homework Statement
Find a piecewise smooth parametric curve to the astroid. The astroid, given by $\phi(\theta) = (cos^3(\theta),sin^3(\theta))$, is not smooth, as we see singular points at 0, pi/2, 3pi/2, and 2pi. However is there a piecewise smooth curve?
Homework Equations
$\phi(\theta) =...
Homework Statement
Find the scalar, vector, and parametric equations of a plane
that has a normal vector n→=(3,−4,6) and passes through the point P(9, 2, –5).
Homework Equations
Ax+By+Cz+D=0
(x,y,z)=(x0,y0,z0)+s(a1,a2,a3)+t(b1,b2,b3)
x=x0+sa1+tb1
y=y0+sa2+tb2
z=z0+sa3+tb3
The Attempt at a...
$\textsf{Find parametric equations .}$
$\textsf{of the line through the point }$
$$P(-3, -4, -2)$$
$\textsf{and perpendicular to the vectors }$
$$u = -6i + 2j + 8k$$and $$v = -7i + 5 j - 2k$$
$\textit{Answer:$\displaystyle x = -44t - 3 , y = -68t - 4, z = -16t - 2 $} $
ok how is this done with...
Homework Statement
Only the second part
Homework Equations
Second derivative:
$$\frac{d^2y}{dx^2}=\frac{d}{dx}\frac{dy}{dx}$$
The Attempt at a Solution
$$dx=(1-2t)\,dt,~~dy=(1-3t^2)\,dt$$
Do i differentiate the differential dt?
$$d^2x=(-2)\,dt^2,~~d^2y=(-6)t\,dt^2$$...
Quick question. I know that if we have a curve defined by ##x=f(t)## and ##y=g(t)##, then the slope of the tangent line is ##\displaystyle \frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}##. I am trying to find the second derivative, which would be ##\displaystyle \frac{d}{dx}\frac{dy}{dx} =...
Homework Statement
Given the ## r(t) = ae^{kt}## , ##θ(t)=kt## find the velocity function that is dependent on ##r##.
##v(r)=?##
Homework Equations
3. The Attempt at a Solution [/B]
My attempt:
1)##r(t) = ae^{kt}##
2)##{\dot r(t)} = ake^{kt}##
From the first equation:
##\ln...
Homework Statement
Given the points P0 = (0,a), P1 = (b,0), P2 = (0,0), write the parametric equation of a circle that intersects the 3 points.
Assume that b > a and both are positive.
Homework Equations
X = h + rcos(t)
Y = k + rsin (t)
r = √((x-h)2 + (y-k)2
Cos (t) = (x-h)/r
Sin (t) =...
Homework Statement
parametric equations are
x=s+2t
y=2s+3t
z=3s+4t
trying to solve for equation of the plane and then take the coefficients of the equation to get the vector normal to the planeHomework EquationsThe Attempt at a Solution
apparently t=2x-y
and s=2y-3x
and therefore the equation...
The derivative for the parametric equations ##x=f(t)## and ##y=g(t)## is given by
##\frac{dy}{dx}=\frac{\Big(\frac{dy}{dt}\Big)}{\Big(\frac{dx}{dt}\Big)}##
The proof of the above formula requires that ##y## be a function of ##x##, as seen in...
Hi all! I have recently taught myself parametrics, and I stumbled upon the butterfly curve. So, I was wondering about some cool equations I can plug into a parametric graphing calculator.
Homework Statement
z = 2x^2 + 5y^2 +2
C is cut by the plane x = 2
Find parametric eqns of the line tangent to C @ P(2, 1, 15)
Homework Equations
z = 5y^2 + 10
dz/dx = 10y
dz/dx (1) = 10
The Attempt at a Solution
z = 10y + 15
y = t + 1
if the slope is 10/1 then delta z = 10 and delta y = 1...
The site http://tutorial.math.lamar.edu/Classes/CalcII/TangentNormalVectors.aspx talks of "the" unit tangent vector of r→(t) = f(t)*i→(t)+g(t)*j→(t)+h(t)*k→(t) as finding "the" tangent vector r'→(t) = f'(t)*i→(t)+g'(t)*j→(t)+h'(t)*k→(t) and normalizing it, and further with finding "the" tangent...
http://tutorial.math.lamar.edu/Classes/CalcII/ParaTangent.aspx
On this page the author makes it very clear that:
$$\frac{dy}{dx} = \frac{\frac{dy}{dt}}{\frac{dx}{dt}}$$
provided ##\frac{dx}{dt} \neq 0##.
In example 4, ##\frac{dx}{dt} = -2t##, which is zero when ##t## is zero. In simplifying...
Homework Statement
Sketch the trajectory over the time interval 0 ≤ t ≤ 10 of the particle whose parametric equations of motion are given by X= t−3sint . And y = 4 − 3cost find the value of x,y,t
Remember that you should be in Radian mode!
Answer
Do you isolate the t first and plug it in...
Hi everybody, i just joined. Here's my first post
I have a few questions.
When looking at curve C and by the eye test, it doesn't pass the VLT, even though x=f (t ) and y=g (t) are functions,...then overall with curve C not passing the VLT, is it still a function since f (t) and g (t) are?
I...
Homework Statement
Sorry to disappoint the math fanatics but no this is not a question that integrates all three topics at once but individual ones. I still need assistance though with the following more so in the reasoning behind them as I feel my logic is flawed...
Homework Statement
Why does
\int_a^b \, y \; dx
become
\int_\alpha^\beta \, g(t) f^\prime(t) \; dt
if x = f(t) and y = g(t) and alpha <= t <= beta?
Homework Equations
Substitution rule?The Attempt at a Solution
I'm not sure how y = y(x) in the integrand turns into g(t). Isn't y a...
How we can know the parametric equation for any curve?
Is there some trick?
Like for parabola ## y^2 =4a x ##
It has general coordinates## (at^2 , 2at) ##
It will satisfy the equation but how in first place we know it?
Also we can have ##(a/t^2, -2a/t) ##, how?
Homework Statement
Find all the points on the following curves that have the given slope:
x=4cost
y=4sint
slope=1/2
Homework EquationsThe Attempt at a Solution
Im not to sure what to do with this question.. I found dy/dx to be -cot(t) but I am not sure if that is even needed for this...
Homework Statement
"Let L1 be the line having parametric equations : x = 2 - s, y = -1 + 2s, z = 1+s and L2 be the line:
x = 1 +t, y = 2+ t, z =2t .
a. Do the lines intersect? If so, find the point of intersection.
b. Find the point P on the graph of L1 that is closest to the graph of L2...
hi,
I'm currently really struggling with an assignment that I've been tasked with.
https://ss1002.files.wordpress.com/2015/01/assignment.pdf
It's mostly theoretical proof questions, which I find difficult. Actual questions I'm fine with.
I have done the first question without issue, as...
So I was watching this video on Khan Academy, and it talks about graphing functions that have values in multiple dimensions. It shows how to represent a linear function in 3 dimensions with a set of vectors, L ={p1 + t(p1-p2)|t∈R} where p1 and p2 are vectors that lie on the line you want to...
Hello everyone, I have another question mark buzzing inside my head.
After the elimination steps of a matrix, I'm having some problems about imagining in 3D.
For example, x=t , y=2t, z=3t what it shows us?
Or, x=t+2, y=t,,z=t ?
Or another examples you can think of. ( Complicated ones of...
Homework Statement
Homework EquationsThe Attempt at a Solution
For part A) my answer was:[/B]
\int_a^b \sqrt{(dx/dt)^2+(dy/dt)^2}dt
The work I used for part A was based off this sites explanation: http://tutorial.math.lamar.edu/Classes/CalcII/ParaArcLength.aspx
For part B)
I simple took...
Homework Statement
I am given a line that passes through the points (x0, y0, z0) and (x`,y`,z`) and a plane in 3D space being defined by these three nonlinear points (x1, y1, z1), (x2, y2, z2), (x3, y3, z3).
Im directed to use cramers rule to find the intersection of the line and plane...
Hi everyone! I'm having some issues with this problem for linear algebra. I understand parametric equations fairly but I'm confused about the unit vector notation
1) Consider the plane r(s,t)=2i + (t-s) j + (1+3s-5t) k find the z component of the point (2,-1, z0)
For what values of s and t is...
Homework Statement
Hi, so confused abou this question that I probably haven't even posted it in the correct section.Here's the question.
A wheel of radius ,r, is situated at the top of a ramp having an angle θ = π/6 rad. At t= 0 the wheel is at rest with its centre at coordinates (0,r) and...
Homework Statement
Consider the curve with parametric equations: x = t - cos t, y = sin t.
Determine exactly the equation of the tangent to the curve at the point where t=-0.5pi.
Homework EquationsThe Attempt at a Solution
The equation of a line is y - y1 = m ( x - x1 )
I substituted t = -pi/2...
Homework Statement
Eliminate the parameter to find a description of the following circles or circular arc's in terms of x and y
and find the center and radius and indicate the positive orientation
x=cos(t) , y = 3sin(t) ; 0< t < pi/2 (should be less than or equal to)
Homework Equations
Not...
i graphed the parametric equations x=t y=t^2 and z=2 on my nspire in 3d and it came out looking like this
(Blush) i have made them pretty so you can see the graph properly
anyways what my question is, is that ( :confused: "is is" ) is the graph supposed to look like that? on a 3d plane? did...
Homework Statement
Find parametric equations for the tangent line to the curve with the given parametric equations at the specified point.
Homework Equations
x = 1+2 \sqrt{t}, \quad y = t^3 - t, \quad z = t^3 + t, \quad (3, 0, 2)
The Attempt at a Solution
I began by...
Hi, so after doing some calculus for projectile motion with air resistance, I obtained two equations of vx(t) and vy(t) that describes the vertical and horizontal motion of the projectile.
Please tell me if I'm wrong, but I believe since both vx and vy are functions of t, can't they be...
sketch the parametric curve and eliminate the parameter to find the cartesian equation of the curve $x=\cos\left({\theta}\right)$, $y=\sec\left({\theta}\right)$, $0\le \theta < \pi/2$
$y=\frac{1}{\cos\left({\theta}\right)}=\frac{1}{x}$
i sketched the curve. how do i do the second part?
Homework Statement
Find the parametric equations for the portion of the parabola y=x^2 from
(-1,1) to (3,9)
Homework Equations
None that I know of.
The Attempt at a Solution
Using knowledge of parametric equations I am not sure how to start. My teacher never went over this...