Calculate fol(3z + 4y) ỉ + (5x + 2y) ) · dř where C is the circular path with center (a, b) and radius m, oriented counterclockwise. Use Green's Theorem. So((3z + 4y) i + (5æ + 2y) 3) · dř =

Q: Find parametrizations for the lines in which the planes 5x - 2y = 11, 4y - 5z = -17

A: Given: The given two planes  5x-2y=11 and 4y-5z = -17 are intersects.

Q: Calculate fo(7x + 6y) i + (4x + 7y) j) dr where C is the circular path with center (a, b) and radius…

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Q: Find parametrizations for the lines in which the planes x + y + z = 1, x + y = 2

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Q: Show by Vector mothads that th ditonce hom fhe fornt P.(Xx, Y,, Z) to the plane Ax -+By+cz-D26 t of…

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Q: Find the trace of the plane 8x – 9y + 2z = -11 in the coordinate xz-plane.

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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: If A = 18zî – 12j + 3yk then find SS,A.ñds where S is the surface of the plane %3D 2x + 3y + 6z = 12…

A: Given, surface S: 2x+3y+6z=12           and                       A→=18zi^-12j^+3yk^

Q: Find parametrizations for the line in which the planes 3r - 6y-2z = 3, and 2x+y-2z = 2 intersect.

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Q: Give a scalar equation for a plane at distance 9 from the plane P with equation -2x-2y+z = 4.

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Q: (a) Prove that the lines with parametric equations: x = 1+t, y = 1+ 6t, z = 2t and x = 1+ 2s, y = 5…

A: aLet the L1 be the line:x=1+t, y=1+6t,z=2tand L2 be the line:x=1+2s, y=5+15s, z=-2+6sThe direction…

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: Is the line x = 4 + 2t, y =-t , z=-4t - 1 parallel to the plane 3x + 2y + z = 7 ?

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Q: (A) Find the Distance between the point (x.y.z) and the plane 3x+2y+6z = 6 If the plane meet with…

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

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Q: Find the distance between the skew lines with parametric equations x = 1 + t, y = 1 + 6t, z = 2t,…

A: Explanation: Given that,  Line 1: x=1+ty=1+6tz=2t Vector 1: 1,6,2 Line 2: x=1+2sy=5+15sz=-2+6s…

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

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Q: Calculate f((5x + 2y)i + (4x + 7y) 3) · dr where C' is the circular path with center (a, b) and…

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Q: Find the point on the plane 3x + 4y +z = 8 that is nearest to (2,0,1).

A: Given the equation of plane:

Q: Draw a sketch of the plane of 4x – 4y + 2z – 9 = 0. -

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Q: find the trace of the plane in the given coordinate plane.  3x + 4z = −2, xy

A: To find the trace of the plane in the given coordinate plane  3x+4z=-2 --- (1) in the xy plane

Q: Calculate Se(5x + 3y) i + (4x + 3y) j) · dĩ where C is the circular path with center (a, b) and…

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Q: Calculate fc((3x + 5y) i + (3x + 4y) j) - dr where C is the circular path with center (a, b) and…

A: According to the greens theorem:

Q: Show that the planes given by 2x − 4y − 3z = 5 and −4x + 8y + 6z = 1 are parallel, and find the…

A: Solution:Given2x − 4y − 3z = 5 −4x + 8y + 6z = 1To prove :The planes are parellel and to find the…

Q: Find an equation of the plane in standard form (ax + by + cz = d) that passes through the point (4,…

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Q: Show that the condition that the curves ..(1) ..(ii) ax? + by =1 %3D a'x² + b'y² = 1 1 1 1 Should…

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Q: Calculate fc((7x + 6y) i + (4 + 7y) j) · dr where C is the circular path with center (a, b) and…

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Q: Find the vector equation that represents the curve of intersection of the paraboloid z = 4x² + y²…

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Q: Which of these describes a parametrized line perpendicular to the plane 3x + 2y + z = 15? A) (2 +…

A: The direction ratio of the normal vector to the plane is (3, 2,1 ) So the perpendicular line to the…

Q: dA, where K is the disk r + (y 2 +?

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Q: Suppose we have: u = and v = . Based on their relative positions in the xy-plane, in which…

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Q: Find the point on the curve of intersection of the paraboloid x² • y² + 2z=4 and the plane x-y+ 2z…

A: This question can be solved by options because we have only two equations and three variables.

Q: Find an equation of the plane through the origin that is parallel to the plane 5x – 4y + 6z + 10 =…

A: Here we have to find the equation of a plane.

Q: Find parametrizations for the line in which the planes 3r- 6y – 2z = 3, and 2x+y-2z = 2 intersect.

A: Given Equation of planes 3x-6y-2z=3 and 2x+y-2z=2

Q: Which of these describes a parametrized line perpendicular to the plane 3x + 2y + z = 15? a)…

A: The direction ratio of the normal vector to the plane is (3, 2, 1) So the direction ratio of the…

Q: Find %3D as if F(x, y, z) = 2xi+2yj+2zk and S is the part of the paraboloid z = 4-x-y lying above…

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Q: Find the points on the hyperboloid 9x^2  - 45y^2 + 5z^2 = 45 where the tangent plane is parallel to…

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Q: The part of the plane 3x + 2y +: - 6 that lies in the first octant

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Q: What kind of conic section is given by the quadratic form ? -11x,? + 84x,x2 + 24x2? = 156 Transform…

A: Given quadratic form is-11x12+84x1x2+24x22=156The above conic is determined and transformed to…

Q: Calculate fc((7x + 6y) i + (4x + 7y) j) dr where C is the circular path with center (a, b) and…

A: The path is given as the circular with center (a, b). Now the integral is: I=∫C (7x+6y)i→ +…

Q: Find the point on the plane 3x + 2y + z = 6 that is nearest the origin.

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Q: Find a plane that contains S and is perpendicular to the plane that contains the points

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Q: Verify that the line r(t) = [1 + t, 1 - t, t] is part of the one sheeted hyperboloid x^2 + y^2 -…

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Q: Find a parametrization for the line segment beginning at P, and ending at P2. P (2,-3,2) and P2(0, –…

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Q: Question 10

A: We have to find equation of plane that is parallel to the plane 6x-5y+8z+18 =0 and  the pale passes…

Q: Give an equation of the plane whose trace in the yz-plane has equation y = 6z, and in the xy-plane…

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Q: Find parametrizations for the lines in which the planes x - 2y + 4z = 2, x + y - 2z = 5

A: Given the planes :- x - 2y + 4z = 2,  x + y - 2z = 5   The Arbitrarily set z =1 and solve the…

Q: Find the vector equation for the line of intersection of the planes x – 5y + 2z = 1 and æ + z = 1 |…

A: To find the vector equation of the line of intersection of given planes.

Q: Suppose that II1 and II2 are parallel planes in R³, given by: II : ajx + biy +cjz = d1, and II2 :…

A: If ∏1: a1x+b1y+c1z=d1 and ∏2: a2x+b2y+c2z=d2 are parallel planes, then, the cross product ∏1×∏2=0.…