A smooth curve is normal to a surface f(x,y,z) = c at a point of intersection if the curve's velocity vector is a nonzero scalar multiple of Vf at the point. Show that the 7 curve r(t) = /ti+ vtj-7(t+3)k is normal to the surface x + y - 7z = 51 when t= 1.

Q: A fluid has density 1200 kg/m3 and flows with velocity ī measured in meters, and the components of ū…

A:

Q: Find the points on the surface xy + yz + zx - x - z2 = 0 where the tangent plane is parallel to the…

A:

Q: Find a parametrization of the vertical line passing through the point (9, 5, 0) using t = z as a…

A:

Q: Find the line integral with respect to arc length (8x + 9y)ds, where C is the line segment in the…

A:

Q: Find the equation for the tangent plane to the surface z=ln(4x2 +3y2 +1) at the point (0,0,0)

A:

Q: Find a point on the ellipsoid x + 3y² + z = 8 where the tangent plane is perpendicular to the line…

A: Introduction: The perpendicular vector to a 3D object f ( x, y, z ) can be calculated by computing…

Q: (4) The equation of the tangent plane to the surface In = z²(x – 2y) + 3z +3 at (4, 2, –1) is 2y

A:

Q: Find the slope of the tangent line at the point (1,−2,−3) on the curve of intersection of the…

A: Given surface is z=5-4x2-y2 The slope of tangent line to the curve that is the intersection of the…

Q: 6. Where does the tangent line to r = at the point (1, 0, 1) intersect the y-z plane.

A: Given:

Q: Find the tangent plane for the level surface f(x,y,z) = 5x3 + cos (5y) + In (6z) = c at the point…

A:

Q: Find a vector tangent to the curve of intersection of the surfaces given by x² + y² + z² = 1 and 2²…

A:

Q: Given the surface z = In(-2x² + 5y² – xy). Find the directional derivative of the function f at…

A:

Q: Find a vector which is tangent to the curve of intersection of the surfaces z = x² – y? and xyz + 30…

A: We will find out the required tangent vector.

Q: Find the equation of the tangent plane to the surface y = 2x2 - z2 at the point (0, –1, –1) %3D | |

A:

Q: The velocity field for a fluid is given by V=(X2t3)i+(2XYT)j. The convective acceleration in the…

A: Given, V=x2t3i^+2xytj^ Here, u=x2t3 and v=2xyt Convective acceleration in y-direction is given by:…

Q: Find the derivative of ƒ(x, y, z) = xyz in the direction of the velocity vector of the helix r(t) =…

A: Concept: A vector is a directed line segment whose length is the magnitude of the vector and with an…

Q: Find an equation of the tangent plane to the surface z=8/2x+4y at the point (2,1,1). z=

A: Please see the explanation below

Q: Let F(x, y, z) =7x2+8y2+13z2. Find the equation to the tangent plane to the level of surface F(x,…

A:

Q: Find an equation of the tangent plane to the surface x = u² + 1, y = v³ + 1, z = u + v at the point…

A:

Q: Find an equation of the tangent plane to the surface z=−2x^2+3y^2−x+3y+1 at the point (1, 1, 4)…

A: We have to find the  equation of tangent plane

Q: 1. Find a vector which is tangent to the curve of intersection of the surfaces z = x2 – y? and xyz +…

A:

Q: Find equations for the tangent plane and the normal line at point Po (X0,Yo,Zo)(1,5,0) on the…

A: As per our guidelines we are supposed to solve only one question. For the solution of the first…

Q: 2. Find the equation of the normal line and the tangent to the parametric surface defined by the…

A:

Q: The gradient vector for the surface xy + xz + yz = 1 at the point (1, 1,0) is (1, 1,0). Select one:…

A:

Q: Compute the vector line integral: x – 4 dx + (x – 4)² + y² -y dy :- 4)2 + y² where C is the circle…

A: To find the below vector integral. ∮C−yx−42+y2dx+x−4x−42+y2dy

Q: find an equation for the plane tangent to thesurface z = ƒ(x, y) at the given point. z = ln (x2 +…

A: Given z=ln(x2+y2)

Q: Determine the equation of the normal line to the surface z2 cos(x + 2y) at the point (2, –1, 0).

A:

Q: Suppose that the plane tangent to z = f(x,y) at (-2,3,4) has equation 4x + 2y + z = 2. Estimate…

A: As the given plane tangent to z=f(x,y) at (-2,3,4) is  4x+2y+z=2, that is z=2-4x-2y thus the…

Q: Compute the directional derivative of f(r.y) = at the point (1, 1) in a direction normal to the…

A: Given that: f(x,y)=x2y at the point (1,1) Ellipse: x2+2y2=1 at the point (1,0)

Q: Compute the directional derivative of f(r. y) = = at the point (1, 1) in a irection normmal to the…

A:

Q: Find the directional derivative of the scalar function of (x, y, z) = xyz in the direction of the…

A: What is Gradient of a Scalar Function: The gradient of a scalar function is vector quantity. It is…

Q: A particle moves along a circular path over a horizontal xy coordinate system, at constant speed. At…

A: Given Motion is a circular motion. r = radius of the circlea = x-cordinate of the centreb =…

Q: A point moves along the intersection of the elliptic paraboloid z = 10x? + 3y² and the plane y = 1.…

A:

Q: Find the line integral with respect to arc length ∫C(6x+4y)ds, where C is the line segment in the…

A:

Q: Find the points on the surface xy + yz + zx – x – 22 = 0 where the tangent plane is parallel to…

A:

Q: Suppose that z is an implicit function of x and y in a neighborhood of the point P = (0, −3, 1) of…

A: Equation of a line passing through point (a, b, c) and parallel to vector <p, q, r> can be…

Q: Find the moments of inertia about the coordinate axes of a thin rectangular plate of constant…

A:

Q: The velocity field for a flow is given by V = ut + vj + wk where u = 3x, y = -2y and w = 2z. Find…

A:

Q: Find the moment of inertia about the x-axis of a thin plate bounded by the parabola x = y - y2 and…

A:

Q: The level curves of the surface z = x2 + y2 are circles in the xy-plane centered at the origin.…

A: Consider , Equation of the surface given:  z = x2 + y2 The level plot is as shown below:

Q: Does the the paraboloid z = 1 – 2x² - - y² have a tangent plane at some – x + y + z = 1?

A: Introduction: Let P0=(x0,y0) is a point in the domain of f, and S is a surface defined by a…

Q: Write parametric equations for the tangent line to the curve of intersection of the surfaces x =…

A: To find: The parametric equations for the tangent line to the curve of the intersection of the…

Q: At what point on the ellipsoid 2x^2 + y^2 + z^2 = 1 is the tangent plane parallel to the plane x + y…

A:

Q: Find a point on the ellipsoid 3x2 + y2 + 3z2 = 1 where the tangent line is parallel to the plane…

A: Given: The equation of ellipsoid is 3x2+y2+3z2=1 and the equation of the plane is -12x+2y+6z=0.…

Q: Determine the equation of the normal line to the surface z2 = cos(x + 2y) at the point (2, –1,0).

A: We have  z2 = cos(x+2y) ⇒cos(x+2y) - z2 = 0  So, F(x,y,z) = cos(x+2y) - z2  Thus,  ∇F(x,y,z) =…

Q: Suppose n is a vector normal to the tangent plane of the surface F(x, y, z) = 0 at a point. How is n…

A: We have asked how a vector n which is normal to the tangent plane of the surface F(x,y,z)=0 at a…

Q: 3. Find the equation of the tangent plane to the surface in R³ defined by z = 2x² + y² at the point…

A:

Q: Find the line integral with respect to arc length (2x + 6y)ds, where C is the line segment in the…

A: The solution of line integral is given as follows :

Q: 14. Find a point on the surface z = 3x- y at which the tangent plane is parallel to the plane 6x +…

A: