This second form is often how we are given equations of planes. Sketch the graph of the parametric equations \(x=\cos^2t\), \(y=\cos t+1\) for \(t\) in \([0,\pi]\). Thus \(y=1-x\). Sketch the graph of the parametric equations \(x=t^2+t\), \(y=t^2-t\). One may have recognized this earlier by manipulating the equation for \(y\): Example \(\PageIndex{8}\): Eliminating the parameter, Eliminate the parameter in \(x=4\cos t+3\), \(y= 2\sin t+1\), We should not try to solve for \(t\) in this situation as the resulting algebra/trig would be messy. y &= \frac{t^2}{t^2 +1} \\ Technology Note: Most graphing utilities can graph functions given in parametric form. The next example demonstrates how such graphs can arrive at the same point more than once. To shift the graph down by 2 units, we wish to decrease each \(y\)-value by 2, so we subtract 2 from the function defining \(y\): \(y = t^2-t-2\). Often this will be written as, ax+by +cz = d a x + b y + c z = d. where d =ax0 +by0 +cz0 d = a x 0 + b y 0 + c z 0. x = s a + t b + c. where a and b are vectors parallel to the plane and c is a point on the plane. Taking derivatives, we have: \[x^\prime = -3\cos^2t\sin t\quad \text{and}\quad y^{\prime} = 3\sin^2t\cos t.\]. We sometimes chose the parameter to accurately model physical behavior. s^2-2s+3 = t^2-2t+3 Solution One method is to solve for \(t\) in one equation and then substitute that value in the second. Solving this system is not trivial but involves only algebra. While the parabola is the same, the curves are different. From the parametricequation for z, we see that we must have 0=-3-t which implies t=-3. We start by computing \(\frac{dy}{dx}\): \(y^{\prime} = 2x\). This, in turn, means that rate of change of \(x\) (and \(y\)) is 0; that is, at that instant, neither \(x\) nor \(y\) is changing. Figure 9.22: Illustrating how to shift graphs in 9.2.3. Contributions were made by Troy Siemers and Dimplekumar Chalishajar of VMI and Brian Heinold of Mount Saint Mary's University. A particle traveling along the parabola according to the given parametric equations comes to rest at \(t=0\), though no sharp point is created.\\. Gregory Hartman (Virginia Military Institute). C Parametric Equations of a Plane Let write vector equation of the plane as: (x,y,z) =(x0,y0,z0)+s(ux,uy,uz )+t(vx,vy,vz) or: s t R z z su tv y y su tv x x su tv z z y y x x ∈ ⎪ ⎩ ⎪ ⎨ ⎧ = + + = + + = + +; , 0 0 0 These are the parametric equations of a line. We again start by making a table of values in Figure 9.21(a), then plot the points \((x,y)\) on the Cartesian plane in Figure 9.21(b). We can quickly verify that \(y^{\prime\prime\prime}=-32\)ft/s\(^2\), the acceleration due to gravity, and that the projectile reaches its maximum at \(t=3\), halfway along its path. A plane is determined by a pointP_0 in the plane anda vector n(calledthe normal vector) orthogonal to the plane. r =(−1,0,2)+s(0,1,−1)+t(1,−2,0); s,t∈R r s t R z s y s t x t x y z s t s t R The straightforward way to accomplish this is simply to add 3 to the function defining \(x\): \(x = t^2+t+3\). This gives \[\cos t = \frac{x-3}{4} \quad \text{and}\quad \sin t=\frac{y-1}{2}.\] The graphs of these functions is given in Figure 9.25. However, other parametrizations can be used. Rather, we solve for \(\cos t\) and \(\sin t\) in each equation, respectively. This gives the point \((-1, 1)\). An object at rest can make a "sharp'' change in direction, whereas moving objects tend to change direction in a "smooth'' fashion. These examples begin to illustrate the powerful nature of parametric equations. The parametric equations limit \(x\) to values in \((0,1]\), thus to produce the same graph we should limit the domain of \(y=1-x\) to the same. We see at \(t=2\) both \(x^\prime\) and \(y^{\prime}\) are 0; thus \(C\) is not smooth at \(t=2\), corresponding to the point \((1,4)\). A curve is piecewise smooth on \(I\) if \(I\) can be partitioned into subintervals where \(C\) is smooth on each subinterval. To determine the equation of a plane in 3D space, a point P and a pair of vectors which form a basis (linearly independent vectors) must be known. This content is copyrighted by a Creative Commons Attribution - Noncommercial (BY-NC) License.
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