12) Prove Ñ' (t) are Smooth space for any always orthogonal (*+Edom R³). that Curve R(t), 7(t) and

Q: cost-sint) √2 r(t) = (3e' cost)i + (3e¹ sint)j + 2k For the space curve below, T= i+ sint+ cost s!).…

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Q: 1) If W = f(x, y,z) is a differentiable and x,y and z are differentiable functions of (t). By using…

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Q: 10. Find the coordinates of the point of intersection of the line L: x = -1+ 3t, y = 2 – 2t, z = 3 +…

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Q: Find the limits of the following vector functions. 2t³-t² tan (2t)\ 21² (i) (ii) lim In 1-0 5t lim…

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Q: a) Sketch the graph of 7(t) =and show the direction of the increasing t, where 0 c) If 7(1)…

A: The vector field is rt=cost, t, sint. Now, if yt=t, then x=cosy and z=siny. Thus, the vector field…

Q: v : {p, q, r, s, t, u} formula (t → g)A (u → p) A (T → s) ^ (tAg → r)A (s t)^ (uAp + 1L) Find an…

A: Given that v:{p,q,r,s,t,u} to {0,1} It satisfies a formula, (t→q)∧(u→p)∧(T→s)∧(t∧q→r)∧(s→t)∧(u∧p→⊥)

Q: Verify the orthogonality condition of each of the equations y" - 2xy' - 2ny = 0 11 ( where n is a…

A: note : Since you have posted multiple questions, we will provide the solution only to the first…

Q: The following r(t) = (3 sint, 4 cost) represents vector-valued function. i) ii) iii) By eliminating…

A: The followingr(t)=(3sint,4 lcost) represents vector-valued function.i) By eliminating parameter 1,…

Q: Find the limits of the following vector functions. -2) 21³-t² tan (21) 21²" (i) (ii) lim In 1-0 5t…

A: Since you have posted multiple questions, we are supposed to answer only first  question. If you…

Q: The normal line to the surface x² + 2y² + ² = 25 at the point (1, -2, 4) is parallel to the vector O…

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Q: Use the inner product (f, 9) = | f(x)g(x) dr %3D in the vector space CU[0, 1] of continuous…

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Q: (5) Consider the plane defined by x+y+z+5= 0. Let (ro, Yo, zo) be an arbitrary point in R³ and let…

A: The equation of the plane is given by:.Let  be an arbitrary point in  and  be the point in the plane…

Q: (a) V₁ ^ V₂ = 0 in ^²(V) iff every alternating bilinear map f: V × V → W has f(V₁, V₂) = 0 (b)…

A: (a) We are given that the exterior power function as follows: Λ2V→V⊗V such that v1 Λ v2→v1⊗v2-v2⊗v1…

Q: 3 find +he equation of Hhe plane perpendicul av bo the plane x+24+5z=|| and passing through the…

A: The given plane is,

Q: 10. Given the curve C defined by the vector-valued function r(t) = a) find an equation of the line…

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Q: (a) Find two points on the line x = -1+3t, y=t + 2, z= 2t – 7. (b) ( Find The point at which the…

A: Equation are given in parametric form: Vary parametric value for different points.

Q: 19. If F(2, y, z) = (2xz, 3y - 22,1 + a2) and C is the path along a2 + y? 1 in the plane z = 0 from…

A: To find: Work done by the field on the particle moving along the loop.

Q: Consider the surface S of equation 2 = x² + y*" An equation for the tangent line tangent to S at…

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Q: G1 is a semi-circle Y: = √4 - X² starting at (2, zero) ends at (-2, zero) while G2 is a directed…

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Q: The divergence and curi of the function f(r,e,p) (r sec 8)e, is ..... O ( (4r] ^ (2) äj6) 4! (4r)…

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Q: What is the equation of the tangent plane to the graph of 3x²+4y²-2²=3 at the point (1,1,2) ? 03 1…

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Q: Let C[0, 1] be the vector space of continuous functions that are defined on the interval [0, 1].…

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Q: Calculate the divergence of cach of the following functions (a) r²î + y²§ + 2²k. (b) yzî + xzj+ xyk.…

A: As per the instruction, we will answer only the first 3 subparts. For (a), For a given vector fx, y,…

Q: X 7. Give a condition on the vector Y 2 such that it lies in Span {[1] 2

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Q: nd meets the uses at A, B and C. Show that the locus of the point of i lanes through A, B and C…

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Q: ) Suppose F(r, y) = x²i + y²j and C is the line segment segment from point P = (0,-5) to Q =…

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