Space CurvesArc length: Find the length of the space curve with vector equationFind vector functions for the intersection of two surfaces:F(1)=(21.1².Given ²(1)= (2x1,1²₁ - 4+²)TNBr(t) =(4t, 3 cost, 3sint) over 0, 27y²x²12 24= 1 and y=x² (= ≥0)44Note: (t² + 2)² =² +4 1² +4Find a unit tangent vector to "Find a unit normal vector to "Find a unit binormal vector to "Velocity, acceleration and curvatureFind the velocity vector, the acceleration vector and the curvature of "Find the tangential and normal components of the acceleration.

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Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter8: Applications Of Trigonometry

Section8.3: Vectors

Problem 60E

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Mass-Spring System The mass in a mass-spring system see figure is pulled downward and then released, causing the system to oscillate according to x(t)=a1sint+a2cost where x is the displacement at time t,a1 and a2 are arbitrary constant, and is a fixed constant. Show that the set of all functions x(t) is a vector space.
(a) Find a vector parallel to the line of intersection of the planes −4x − 2y + z = −7 and 5x +2y − 5z = 12v→ = (b) Show that the point ()(1, 1, −1) lies on both planes. Then find a vector parametric equation for the line of intersection. r→ =
1. A vector-valued function r(t) in R3 traces a circle that lies completely on the plane z=3 and with center (0,0,3). (1.1) Find the component functions of r(t). (1.2) Find the equation in vector form of the tangent line to r(t) at the point where t=π.
Sketch the space curve represented by the intersection of the surfaces z = x2 + y2, y = 2. Then use the parameter x = t to find a vector-valued function for the space curve
1. Find the domain of the vector function. (Enter your answer using interval notation.) r(t) = 16-t2, e−2t, ln(t + 3) 2. Find a vector equation and parametric equations for the line segment that joins P to Q. P(3.5, −1.4, 3.1), Q(1.8, 0.3, 3.1) 3. At what points does the curve r(t) = t i + (5t − t2) k intersect the paraboloid z = x2 + y2?
Streamlines and equipotential lines Assume that on ℝ2, the vectorfield F = ⟨ƒ, g⟩ has a potential function φ such that ƒ = φxand g = φy, and it has a stream function ψ such that ƒ = ψy andg = -ψx. Show that the equipotential curves (level curves of φ)and the streamlines (level curves of ψ) are everywhere orthogonal.
The mass in a mass-spring system (see figure) is pulled downward and then released, causing the system to oscillate according to x(t) = a1 sin ωt + a2 cos ωt where x is the displacement at time t, a1 and a2 are arbitrary constants, and ω is a fixed constant. Show that the set of all functions x(t) is a vector space.
Find a vector parametrization of the curve x=−2z^2 in the xz-plane. Use t as the parameter in your answer
Bendixson’s criterion The streamlines of a planar fluid floware the smooth curves traced by the fluid’s individual particles.The vectors F = M(x, y)i + N(x, y)j of the flow’s velocity fieldare the tangent vectors of the streamlines. Show that if the flowtakes place over a simply connected region R (no holes or missingpoints) and that if Mx + Ny 0 throughout R, then none of thestreamlines in R is closed. In other words, no particle of fluid everhas a closed trajectory in R. The criterion Mx + Ny ≠ 0 is calledBendixson’s criterion for the nonexistence of closed trajectories.
How do you find the length of a space curve given by a vector function r(t)
Dierentiable curves with zero torsion lie in planes That a sufficiently di¡erentiable curve with zero torsion lies in a plane is a special case of the fact that a particle whose velocity remains perpendicular to a fixed vector C moves in a plane perpendicular to C. This, in turn, can be viewed as the following result. Suppose r(t) = ƒ(t)i + g(t)j + h(t)k is twice di¡erentiable for all t in an interval 3a, b4 , that r = 0 when t = a, and that v # k = 0 for all t in 3a, b4 . Show that h(t) = 0 for all t in 3a, b4 . (Hint: Start with a = d2r/dt2 and apply the initial conditions in reverse order.)
(a) Find the y-intercept of the line in 2- space that is represented by the vector equation r = (3 + 2t)i + 5tj. (b) Find the coordinates of the point where the line r = ti + (1 + 2t)j −3tk intersects the plane. 3x −y−z = 2.

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Arc length: Find the length of the space curve with vector equation r(t) =(4, 3cost, 3 sint ) over [ 0, 2,7]