Sketch the plane curve. r(t) = t?i + 2tk, [0, 2]

Q: Find T, N, and B for the curve r(t) = (10t, 10t, e') at (0,0, 1).

A: Given: we have ,

Q: Which of the following is the equation of the plane that is tangent to the surface 2 x² + y? – x z –…

A: Consider the provided information, The provided surface, 2x2+y2-xz-3z=1 Find the partial derivative…

Q: In what plane does the curve r(t) = t i + t2 k lie?

A: r(t) = t i + t2 k We need to find the plane of the curve located in?

Q: Find an equation of the tangent plane to the surface z = 5y – 4x? at the point (-4, 2, -44).

A: The solution is given as

Q: Find the coordinates of the point where the line F = (1+ 2t)i + tj – 2tk intersects the plane 2x – y…

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Q: Determine A so that the curve y = 4x+ 16 can be written in parametric form as x(t) = t – 5, y(t) At…

A: Put value in given equation and simplify

Q: Let r(t) =(t, 4/3 t3/2, t2) Find the equation of the osculating plane at the point correspondingto t…

A: Given that, rt=t, 43t32, t2rt=t, 2t33, t2

Q: Sketch the plane curve. r(t) = 4ti – tj, [0, 4] %3! y y 4 4 2 X 10 15 10 15 -2 -2 y y 4 4 2 10 15 10…

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Q: Sketch the plane curve r(t) = 3ti − tj and find its length over the given interval [0, 3] .

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Q: Find the equation of the tangent plane to +y? at the point (2,-2,5) the surface z= 4 O r + 4y - % =…

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Q: Evaluate | F. dR, where F(x, y, z) = (x, –2, y) and C is the curve Ř(t) = (2t, 3t, –t²), 0 <t < 2.

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Q: Use Eq. (5) to find the point P nearest to Q = (2, 1, 2) on the plane x+y+z=l.

A: Consider the provided information, Find the point P. The normal vector to the plane is 1,1,1. And…

Q: Find the length of the curve. r(t) = ( 8t, t², ), 0 <t< 1 %3D 12

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Q: Find T, N, and k for the plane curve.r(t) = (2t + 3)i + (5 - t^2)j

A: Consider: rt=2t+3i+5-t2j. Find T, N, k. We have, if r=ui+vj then r'=u'i+v'j and r=u2+v2.

Q: find the trace of the plane in the given coordinate plane.  −x + y = 4, xz

A: Find trace of the plane in the coordinate plane -x+y=4 in xz

Q: Determine the distance between: the pa rallel line x-a++ X - 3+S+t t9= 5-t,tER and the…

A: Given line is                                                     x=2+ty=5-tz=-4 point on the line…

Q: Sketch the plane curve. r(t) = 2ti – tj, [0, 2] y y 2. 1 1 -1 1 4 -1 1 4 5 -1 y -1 2 3 4 1 2 3 4 5…

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Q: Sketch the plane curve r(t) = t3i + t2j and find its length over the given interval [0, 1] .

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Q: Find an equation of the tangent plane (in the variables x, y and z) to the parametric surface.…

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Q: nd the tangent plane to the equation z = a? – 4y? – 2y at the point ( – 3, 3, – 33)

A: In this question, concept of tangent  is used. Tangent A tangent is the line that runs parallel to…

Q: In three space, find the parametric equations of the curve C of intersection of the cylinder S: x2 +…

A: Here we have to find the parametric equations of the curve C of intersection of the cylinder S:x2+y2…

Q: Which of the following is the equation of the plane that is tangent to the surface 2 x2 + y? x z –…

A: Given: 2x2+y2-xz-3z=1

Q: Sketch the curve r(t)=f°i+tj and indicate the orientation. Find r'(t) and sketch r(1) & r°(1).

A: First we will plot the given parameters.

Q: Find the length of the curve. r(t) = 2i + t2j + t³ k, 0 sts 1 %3D

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Q: 3. Find the equations of the normal plane and the osculating plane to the curve at the given point…

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Q: Find the barycentric coordinates of p with respect to the affinely independent set of points that…

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Q: Find an equation of the tangent plane to the surface z = 5y – 4x? at the point (4, 2, -44).

A: Given: Equation of surface is z=5y2-4x2 and a point -4, 2, -44. To find: Equation of tangent plane…

Q: Q.1 Find the equation of the plane that is orthogonal to line L:(t – 1,2 –,t/2) at (t =2 ).

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Q: Let C be the curve r(t) = (2t, t², t°/3). (a) Compute the Frenet frame for C. (b) Calculate the…

A: The position vector is rt=2t,  t2, t33. Differentiate with t and obtain the unit tangent vector:…

Q: Find an equation for the projection of the curve on yz plane. O y + z? = 3 O y? + z? = 4 O y? + z? =…

A: For projection in yz plane equation must be in terms of y and z only. Here x=t y=3 cos t z=3 sin t

Q: Evaluate [ f (x,y,z) ds given that f(x,y,z) =, and the curve C is ř(t) = 2ti +t²j+÷tk (1<t<∞) %3D…

A: Given: f(x,y,z)=x2xy+6yz r→(t)=2t i→+t2 j→+13t3 k→   1≤t&lt;∞

Q: Let C2 be the curve defined by R(t) = (t² +t + 3, 4t – 5, 2+ t – t2), t e [0, 1]. %3D | (x + y+ 2)…

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Q: Find the point at which the line F (t) = +t intersects the yz plane.

A: Given: An equation of a line in the parametric form is, r→t=&lt;2,-7,7&gt;+t&lt;-5,-6,6&gt; The…

Q: Let r(t)=(t, (4/3)t^(3/2), t^2). Find the equation of the osculating plane at the point…

A: Let the given point be (x0, y0, z0), to find the osculating plane find the cross product of the…

Q: ). Find a parametric representation for the line of intersection of the following two planes: I - y…

A: We use cross product to find the direct vector for the line.

Q: Find the length of the curve r(t) (1,t²,), 0sts1 %3D «3 Ost<1

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Q: Find equations for the osculating, normal, and rectifying planes of the curve r(t) = t i + t2 j + t3…

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Q: Find T, N and K for the plane curve: r(t) = ti + Incos tj

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Q: Find the parametric equation for the line where the planes 7x – 6y + z = 10 and 2x + y – 3z = 4…

A: We have, r(t) = (0, -2, -2) + t(17, 23, 19) The detailed solution is as follows below:

Q: Show that the curve al lies in plane and b) finel the equation of the plane that contains the curve…

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Q: Find the length of the parametric curve x = 2t + 1, y = 3 + 2tž, 0 < t <%.

A: Find the Length of the curve

Q: 2. Sketch the plane curve by plotting points then indicate the orientation of the curve. x = t2 – 4…

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Q: Find the projection of v = onto the line l of R³ given by the parametric equation l = tü, where i…

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Q: Find the point at which the line 7(t) +t intersects the | xz plane.

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Q: Sketch the plane curve. r(t) = 3ti - tj, [0, 3] y O 2 1 -1 -2 y 3 2 - -1 -2 2 4 4 Find its length…

A: This question is related to vectors and parametric equations, we will solve it using given…

Q: Complete the parametric equations for the line wherekche planes 2=0and-2z + 2y + 2z -4 intersect.…

A: The equation of the planes are: P1: 2z = 0 ⇒ z = 0                                  .........1 P2:…

Q: Let Q = 4x2 + 6xy+ 4y? . The points on the curve Q = 7 which are closest to the origin are: %3D %3D…

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Q: A Show that the curve F(t) = lies in a plane.

A: