sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter. Surface:      x2+y2=4;     z=x2  Parameter:    x=2sin(t)

Q: Find an equation of the tangent plane to the given parametric surface at the specified point. r(u,v)…

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Q: a) Find a vector parametric equation for the ellipse that lies on the plane y – 4x + z = -8 and…

A: (a) Given y-4x+z=−8 and x2+y2=4 Let x=ucosv and y=usinv Then z=−8+4x−y ⇒z=−8+4ucosv−usinv Hence,…

Q: The tangent plane to the surface x? + yz +1 = 0 at the point (x, y, z) (2,–5, 1) passes through…

A: The given problem is to find through which point the tangent plane to the surface at given point, we…

Q: 41. The cone z = /x² + y² and the plane z = 1 + y

A: Solution: The objective is to find a vector function that represents the curve of intersection of…

Q: The length of a curve given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in the…

A: Given information: Parametric equation of curve is : x=x(t)y=y(t)z=z(t) We need to find the length…

Q: A vector normal to the surface y-:=3 at the point (-1-2, 2) is (a) i+2j+k (b) -i+j+2k) (c) (21 -j+k)…

A: Definition used -              For finding a normal vector to the given surface, We have to find…

Q: (a) Find a vector parametric equation for the part of the saddle z = xy inside the cylinder x² + y²…

A: The equation of surface is defined by the part of the saddle z=xy inside the cylinder x2+y2=36.

Q: Check that the point (−1,−1,1) lies on the given surface. Then, viewing the surface as a level…

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Q: Determine a scalar equation for the plane that passes through the point (2, 0, -1) and is…

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

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Q: How do you find the length of a space curve given by a vector function r(t)

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Q: Find the rectangular equation for the surface by eliminating the parameters from the vector-valued…

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Q: Check that the point (1,1,3) lies on the given surface. Then, viewing the surface as a level surface…

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Q: Consider the line perpendicular to the surface z = x2 + y at the point where x = 1 and y = -1. Find…

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Q: Use a parameterization to find the flux F•n do of F = zʻi+ xj - 2zk in the outward direction (normal…

A: Given : A Parabolic cylinder z = 1 - y² by the planes x = 0, x = 1, and z = 0

Q: Evaluate Curlv-ñ, where v = 2xyi + (x-2y)j+xzk and ñ is a unit vector normal to the surface shown in…

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Q: Find the perpen dicular trajec ories of the plane faimily its intersection X -y +2 = Ł Cヒ cons tan+)…

A: Given surface is x -y +z = k  --- (1)                          ⇒z = k -x+y The parametric equation…

Q: Find a vector parametrization of the curve x=−2z^2 in the xz-plane. Use t as the parameter in your…

A: Find a vector parametrization of the curve x=−2z^2 in the xz-plane. Use t as the parameter

Q: Evaluate Curlv•ñ , where v = 2xyi + (x² – 2 y)] + xzk and ñ is a unit vector normal to the surface…

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Q: Find a parametrization of the sphere x2 + y2 + z2 = a2.

A: we have to find a  parametrization of the sphere x2 + y2 + z2 = a2.

Q: Evaluate Curlv.ñ , where v = 2xyi +(x² – 2y)j + xzk and ñ is a unit vector normal to the surface…

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Q: Find the vector and parametric equations for the line through the point P(1, 1, -3) and orthogonal…

A: Given: Point= (1, 1, -3) Orthogonal to plane: 3y-4x+2z=3 The direction vector for this line is v=…

Q: Verity Gauss' theorem for the vector feld f (x,y, z)= (y,x,2) on the surface of the sechere with…

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Q: Use a parameterization to find the flux do of F = z²i+ xj – 4zk in the outward direction (normal…

A: Apply Divergence theorem to the given surface integral, ∬F.ndσ=∫∫∫DivFdV

Q: Assume that an object is moving along a parametric curve and the three vector function T (t), N(t) ,…

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Q: Find the flux of F = z²k upward through the part of the sphere æ² + y² + z² = a² in the first octant…

A: Given vector field is  F=z2k to find the flux through sphere x2+y2+z2=a2 in the first octant.

Q: Find a parametrization of the line in which the planes x- 2y + 2z = - 1 and – 4x+ 3y – 3z = -1…

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Q: Use a parameterization to find the flux F•n do of F = z'i+xj- 4zk in the outward direction (normal…

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

A: Given data: Equation of cone is y=2x2+9z2. Equation of plane is y=5+3z.

Q: Find a vector tangent to the curve of intersection of the two cyclinders ? + y² = 98 and y² + z² =…

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Q: Write an equation for the plane tangent to the surface F(x, y, z) = 0 at the point (a, b, c).

A: Given surface is Fx, y, z=0 and the point is a, b, c.

Q: Find a parametric representation for the surface: The plane through the origin that contains the…

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Q: 6. (a) Show that the curve represented by the vector function F(t) = (5t, 3t – 4, 2t² – 6) is at the…

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Q: Find a vector parameterization of the plane −5x−4y−5z=18. r(s,t)=_____? (Use s and t for the…

A: A parametrization for a plane can be written as                                       r = s a + t b…

Q: Use a parameterization to find the flux F•n do of F = z'i+xj-3zk in the outward direction (normal…

A: We have to fin the flux ∬SF⋅ndσ, where F=z2i+xj−3zk. The intersection of the parabolic cylinder…

Q: Find the vector and parametric equations for the line through the point P(-3,0, – 3) and orthogonal…

A: Given data is:point P-3,0,-3 and plane -1x+1y-5z=1

Q: Select the correct parametrization of the following curve: The intersection of the plane y = 2 with…

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Q: Sketch the space curve represented by the intersection of the surfaces. Surfaces Parameter x2 + y2 +…

A: Given surfaces are, x2+y2+z2=4x+z=2 The parameter is, x=1+sint The sketch of the space curve is…

Q: Find the flux of F = z²k upward through the part of the sphere x² + y² + z² = a² in the first octant…

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

A: The two planes are x + y + z = 2 and x + y = 6

Q: Consider the curve r (t) = (t, t³, t®); t>0 The curve intersects the plane (in a three dimensional…

A: Topic- intersection point of plane and curve

Q: Consider the space curve with vector equation r(t) = . Find (i) symmetric equations for the tangent…

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Q: 1) Find a vector function r(t) that represents the intersection of the surfaces: = /x² + y² z = y +…

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Q: Check that the point (1, –1, 2) lies on the given surface. Then, viewing the surface as a level…

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Q: Exercise 7.2.8. Apply Stoke's theorem to find F - dr, where F = yi+zj+ækover the surface S, %3D…

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Q: QA-evaluate the flux of the filed F-xi+yj+zk through the upper part of the cylinder x² + z² = 1 that…

A: Given

Q: See attached photo.  Please provide the flux.

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Q: 4. (8 points) Find a vector parametrization for the intersection of the plane x +y+z = 3 and the…

A: Use polar coordinates for Parameterization

Vector Arithmetic

Vectors are those objects which have a magnitude along with the direction. In vector arithmetic, we will see how arithmetic operators like addition and multiplication are used on any two vectors. Arithmetic in basic means dealing with numbers. Here, magnitude means the length or the size of an object. The notation used is the arrow over the head of the vector indicating its direction.

Vector Calculus

Vector calculus is an important branch of mathematics and it relates two important branches of mathematics namely vector and calculus.