Q1. a) Let T:R → R² be defined by T(x, y, z)=(x+y,y-z). Is T invertible? Justify your answer. b) Let T:M -→R be defined by T(A)=det(A). Is T linear? Explain your answer. c) Let u = (0,-4,3). Determine all vectors orthogonal (perpendicular) to u.

Q: Prove that if u is the vector function with the constant value C, then du/dt = 0.

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Q: Find the work done by a force F = 6ỉ + 5j– 3k when a particle moves from point P(-3, 4, 5) to point…

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A: We find the flux of the vector field F(x, y, z) = [9y, 2xy, 4yz + 234xy] through the unit cube [0,1]…

Q: Compute feF. dr where C is the curve of intersection of the paraboloid z=9-x² - y² and the plane z =…

A: To compute the line integral ∫CF⋅dr. We have F=2x2+3zi+4ey+8xj+6z2+7yk and  C is the curve of…

Q: Consider the vector function r(t)=⟨t,t^2,t^4⟩ Compute r′(t)=⟨  ,  ,  ⟩ T(1)=⟨  ,  ,  ⟩ r′′(t)=⟨  ,…

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Q: Prove that u is orthogonal to v - proju(v) for all vectors u and v in Rn, where u ≠ 0.

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Q: Verify Stoke's theorem for the vector point function yz2 (xz² .3 + ex over the surface S of the cube…

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Q: Let t e R. Discuss the lincar independence of the vectors over R: aj %3D(1, 1, 0), аз %3 (1, 3, —1),…

A: we will use the connection of linear independence of a set and columns of an Invertible mat

Q: The divergence of the vector F = a²yî + æyz} – a*y°k is x2 y? k is -

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Q: Find the flux of the vector fleldF = (x – y*)i+ 4yj + x'k out of the rectangular solid [0, 1] x [1,…

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Q: Find the flux of the vector field F = (y, – z, x) - 2, - across the part of the plane z = 2 + 3x +…

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Q: if A=i+j+X, B=zi-j+3ksfind vector normal to B and in plane of A and B- ans. $2(3i+9j+x)

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Q: Let o = 2 In(r), where r= V x2 + y2. Express Vo in terms of the unit radial vector e, in R2. tep 1…

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Q: Compute the curl, V x F, of the vector field. F(x, y, z) = 7xi + 9yj + 2zk curl(F) = -7xi – 5yj +…

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Q: for the vector field F = 2xz i+ (1 – 4xy² ) j led by the plane z = 6 – 2x² – 2y² and the plane

A: We have to find the following conditions according to the given information

Q: f f(x, y, z) = xy: on of vectorî - %3D

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Q: Compute the vector assigned to the pointP = (−3, 5) by the vectorfield:(a) F(x, y) = (xy,y − x)(b)…

A: To Determine : (a) F(x, y) = (xy,y − x)(b) F(x, y) = (4, 8)(c) F(x, y) = 3x+y , log2(x + y) Given:…

Q: Find the spacing and involution for the following vector functions 1- ƒ(x, y, z) = (xi + yj + zk]…

A: Topic:- vector

Q: 3. The work Wdone by = xf + 3xyj - (x + z)k to move a particle from point 1 to point 2 is defined by…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Q1:- Find the domain of the following vector fumctions:- (a) f(t) = (cos nt)i - Ln(t)j + Vt-2k (b)…

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Q: Q2] (A) Find the surface area of the portion of the half ball with radius a which cuts off by the…

A: (A)Consider the given information.Let the ball is defined by the sphere and the equation with radius…

Q: Find the work done by the force F- Tyi + (y - ±)j over the straight line from (0,0) to (3,6).

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Q: Let, T:R' → R;T(x, y, z) = (x + y+ z,2y + z,2 y+3z) verify that [T][v]s = [T (v)]; for any vector v…

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Q: Let x 0 and y 0. (a) If xly, show that {x, y} is a linearly independ- ent set. (b) Extend the result…

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Q: Find all points of intersection between the vector functionr(t) =⟨t, t2, t3⟩ and the plane defined…

A: Since the vector is contained in the plane, so all the points of the line intersect with the plane.…

Q: Find the Vector function that represents the curve of the intersection of the two surfaces.  The…

A: Given equations are x2+y2+4z2=4 ..... (1) and x2+z2=1 ..... (2) From equation (1) z2=(4-x2-y2)/4…

Q: 3. Use the Wronskian to determine whether the functions fi = e*,f2 sinx and f3 = x are linearly…

A: To find- Use the Wronskain to determine the functions f1 = ex, f2 = sinx and f3 = x3 are linearly…

Q: What is the most general solution to u + v + w = 1, u − w = 2 in vector form?

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Q: Find the gradient vector field (F(r, Y, 2)) of f(x, y, z) = In(4x + 6y + 22) . F(7, y, 2) = (

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Q: Let, T :R → R';T(x,y,z) =(x+y+ z,2y+ z,2y +3z) 7. verify that [T][v]s = [T (v)], for any vector v in…

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Q: Find the flux of the vector field F = (y, – z, x) across the 2 + 4x + 3y above the rectangle part of…

A: “Since you have asked multiple question, we will solve the first question for you. If youwant any…

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Q: Find the directional deriu atrue Du f(-3,-1,3) for the function f (xiy,z) =xy + yz +Z X in the…

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Q: Find a vector parallel to the plane 2x - y - z = 4 and orthogonal to i + j + k

A: Given:A plane 2x - y - z = 4 orthogonal to i + j + k

Q: Let u= and w- Find the unique vector v in R'such that u.v 1, v is orthogonal to w, l|v|| V3, and y>…

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Q: Find the divergence of the vector field. V(x, y, z) = 7e*Yj – 9e*Yj + 8EY²K - div(V) II

A: A detailed solution is given below.

Q: Evaluate the line integral F*dr, where F(x,y,z) = (3sin(x), -4cos(y), 4xz) and C is given by the…

A: Find your answer below

Q: Exercise 6 Is the set of vectors V = {i² + 2t – 1,3t + 5, -t² + t + 6} linearly dependent?

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Q: In Rn with inner product{x, y} = xTyderive a formula for the distance between two vectorsx = (x1, .…

A: Take two vectors x and y in an inner product space x=x1,…,xn and y=y1,…,yn then x,y=xTy

Q: Let the vector ū = (k,5,–1) and v = (2k,k,3) are perpendicular in R', find the value of at related…

A: Given: u→=k,5,-1v→=2k,k,3

Q: Find a vector tangent to the curve of intersection of the two cyclinders a2 + y? = 8 and y² + 22 = 8…

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Q: Find the Duf of f(x, y,z) = z In(x+ y) + e3y on the vector u = 3i -j-4k at Po(1,0,1)

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Q: (b) Solve the electric flux through a paraboloid z = x² + y² – 16 that lies below plane %3D z = 0.…

A: As per our guidelines, we can only solve first question and rest can be reposted. The solution for…

Q: 4. Determine whether the vector functions are linearly independent. Show your work. 1 4². 2t 8t 2

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Q: Q1/A/ Show that in plane the non zero vector N = ai+ bj is:- perpendicular to the line ax +by +c =0…

A: We have to show that the vector N=ai+bj is perpendicular to the line ax+by+c=0. We have to find the…

Q: Let, T:R' → R';T(x, y,z) = (x – y,2x + 3y + 2z,x+ y+2z) . verify that [T][v], = [T (v)]; for any…

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Q: Show that (x, y) = 4x1y1-7x2Y2 for vectors x = (x1, x2) and y = (V1 Y2) in R2 defines an inner…

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.