Find the orthogonal trajectories on the sphere x2 + y2 + z2 = a2 of its intersection with the paraboloids xy = Cz

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: 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: Find the area of the surface, which is part of the surface z = 1 + 3x + 2y2 that lies above the…

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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: Determine the point on the plane 2x + 3y − 4z = 0 that is closest to the point P(1, 1, 1)

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

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Q: Use LaGrange multipliers to find the points on the surface 2² = x² + y² which are closest to (4, 2,…

A: Given surface z2=x2+y2 and the point (4,2,0). The objective is to find the points on the surface…

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: Evaluate SdV where G is enclosed by the paraboloids z = G z = 4 x² - y². Answer: 2π 3x² + 3y² and

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Q: Use Lagrange's multiplier(s) to find the maximum and minimum for f(x, y, z) = x² + 2x + 2y2 + 3z2 on…

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Q: 23. Find the coordinate vector [r]B of x relative to the given basis B = {b1, ...,bn}. 8 bị by 4 b3…

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Q: Determine the scalar equation of the plane that contains A(-1, 3, 1) and is perpendicular to both 5x…

A: The normal vector of a plane is a vector that is perpendicular to the plane. The coefficients of x,…

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: Find the points on the surface 1x² + 1y² + 3x² = 1 at which the tangent plane is parallel to the…

A: We know that two planes are parallel if their normal vector are parallel. That is their, gradient…

Q: Let C be the positively oriented square with vertices (0, 0), (3, 0), (3, 3), (0, 3). Use Theorem to…

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Q: Find the scalar equation for the plane passing through the points P1(4, -4, 5), P₂(1, 1, 10), and…

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Q: The upper half of the unit sphere x2+y2+z2=1 is z=Square root of 1-x2-y2. Find its centroid

A: Solution Given The intersection gives z2=1-x2+y2we need to find the centroid

Q: Find the point on the plane 3x - 2y + z = 16 that is closest to the point (1,2,3). (Use the method…

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Q: Find the distance from the point (1,1,1) to the plane 2x+2y+z=0

A: Solution :-

Q: 285. Find the scalar equation of the plane that passes through point P(-1, 2, 1) and is…

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Q: Find a parametrization for the line SEGMENT between (- 5, 7, 6) and (2, – 6, – 9). |

A: Parameterizing a line segment, which joins the points (x1,y1,z1) and (x2,y2,z2), helps in obtaining…