1. Find a number a so that the change of coordinates s =r+ ay,t = y transforms the integral SR drdy over the parallelogram R in the ry-plane into an integral |a(1, y) dsdt | a(s,t)| over a rectangle T in the s-t plane, where the parallelogram R has vertices (0,0), (10,0), (12,3), (22, 3).

Q: Let Z = (1, 1, 2) be a point in R³. Find the closest point to Z tha lies on the plane give by -x+y-z…

A: The normal vector to the plane is <-1, 1, -1>. The point you seek would have to be some…

Q: Q.2)(A) Find a formula for the area of the triangle in the xy-plane with vertices at A(0, 0), B(a1,…

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Q: Show that the points A = (1, 2), B = (1,0), and C = collinear and express P = (-7,0) in barycentric…

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Q: 20x + 20y = 80X20-3 3. Rajah di sebelah menunjukkan susunan gambar di dalam sebuah bingkai gambar.…

A: The distance between the pictures is 2 cm and the distance between the picture and the edge of the…

Q: Find a parametrization of the line in which the planes x- 2y - 2z = 3 and -3x + 3y + 3z = -6…

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Q: Find the scalar equation of the plane that contain the line 7 = [2,1,7] + t[0, 1, 0], t e R and is…

A: Here we will find out the required plane.

Q: Let L be the line passing through the point P(−4, −3, 4) with direction vector d=[-3, 3, 0]T, and…

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Q: Let C be the positively oriented square with vertices (0, 0), (2, 0), (2, 2), (0, 2). Use Green's…

A: Introduction: According to Green's Theorem, a line integral of a vector over a closed curve C…

Q: Find a plane through P (2, 1, 1) and perpendicular to the line of intersec- tion of the planes:…

A: find a plane through P2,1,1 and perpendicular to the line of intersection of the planes  2x+y-z=3 ,…

Q: he vector v = , is tangent to the surface x2 + 2y3 - 3z2 = 3 at the point (2, 1, 1).  Find a.

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Q: Find the scalar equation for the plane passing through the point P=(−5, 3, 2) and containing the…

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Q: Find the trace of the plane 3x − 2y + 5z = 4 in the xy-plane.

A: The given equation of the plane is 3x-2y+5z=4. The trace of the plane in the xy-plane is the…

Q: Find the value of k so that the line [x, y, z] = [2, -2, 0] + t[3, 5, k] is parallel to the plane 4x…

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Q: Let 5x + 3y + 10% = 0 be the equation of the 3 plane in R³ spanned by (1, a, 0) and (1,0, b). Find…

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Q: 4. Consider the vector function r(z, y) = (x, y, x² + 2y*). (a) Re-write this vector function as…

A: For the given vector function, PART A The surface function is given by, f(x,y)=xi^+yj^+(x2+2y2)k^…

Q: Find the plane through P = (4, −1, 9) containing the line r(t) =〈1, 4, −3〉+ t〈2, 1, 1〉.

A: The plane contains the line r(t)=1, 4, -3+ t2, 1, 1, so a vector in the plane is 2, 1, 1. The line…

Q: Find the distance between the line 2x - 3y + z = 6, 2y- z = 4, and the plane passing through the…

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Q: Vector A = 6i + 2j - 3k Find the

A: Given: Vector A=6i+2j-3k  To find: Scalar component of A perpendicular to the x-axis.

Q: 2. Find a vector that is perpendicular to the surface x² + y² + z² = xyz + 8 at the point (1,2,3).

A: Given that x²+y²+z²=xyz+8 At (1,2,3)

Q: Show that the line x + 1 =y/−1=(z − 2)/2 is in the plane 2x + 4y + z = 0.

A: Given equation of a line, x+1=y-1=z-22 And equation of a plane, 2x+4y+z=0

Q: Consider the three following points of R'. A = (0, 2, 2) B = (1, 1, 1) C = (-1, 1,2) (a) Find the…

A: The scalar equation of a plane is of the form: ax+by+cz+d=0, where a, b, c ,d are constants. The…