Show that if r(t) is perpendicular to the tangent vector r′(t) for all t, then the curve r(t) lies on a sphere with the center the origin.

Q: Verify Stokes's theorem for F =, where C is the curve C: x² + y² = 4, z = 2

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Q: Find an equation of the tangent plane to the surface g(x, y) = x2 + y2, at the given point (1, −1,…

A: We have given the surface; g(x, y) = x2 + y2 and the point (1, −1, 2)

Q: r(u, v)=(w+v)i(u-v)j+v²k at the point P(2, 0, 1)

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Q: Find the equation of the tangent plane at the point T(1,1,4) of the sphere (x – 1)2 + (y – 2)² + (z…

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Q: find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the…

A: Solve the following

Q: Find the equation of the tangent plane to the surface z=9x^5+2y^5+1xy at the point (0,−1,−2) z=

A: From tangent plane equation to a curve

Q: Find an equation of the tangent plane to the surface at the given point. h(x, y) = In Vx2 + y², (6,…

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Q: Find an equation of the tangent plane to the surface at the given point. h(x, y) = In Vx2+ y², (12,…

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Q: Find the equation of the plane tangent to the surface f(x,y) = x In(x y) at the point (1, e).

A: We will find the equation of tangent plane to the given surface.

Q: Find the point(s) on the surface z = 5xy, at which the tangent plane is horizontal.

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Q: Show that this curve y represents the vector (1,0) as v = 1- +0- in the sense that, for an arbitrary…

A: Given that γ(t)=(t,t2)in R2, t∈[-1,1] The objective is to show that curve represents the vector…

Q: Find the unit tangent vector T(t) and a set of parametric equations for the tangent line  r(t) =…

A: The given space curve rt=e2ti+cost j - sin3t k. Its derivative is r't=2e2ti-sint j - 3cos3t k. At…

Q: Consider the curve defined as r(t) = e^ti + e^tj + t k Determine the point on the curve where the…

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Q: A space curve is given by the vector function r(t) = (3t, t² + 6t, t³ ). Then the unit tangent…

A: GIVEN: r(t) = {3t, t2 + 6t, t3}

Q: Find the curvature for a circle of radius a > 0, centered at the origin, and traced cut by the…

A: Explained below

Q: Find the unit tangent vector T(t) and find a set of parametric equations for the line tangent to the…

A: Given the curve     rt=2sin(t),2cos(t),4sin2(t) and the point P1,3,1.

Q: If a curve has the property that the position vector r(t) is always perpendicular to the tangent…

A: given that, a curve has the property that the position vector r(t) is always perpendicular to the…

Q: Find an equation of the tangent plane to the surface at the given point. h(x, y) = In Vx2 + y²,…

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Q: Let R be a vector function whose graph is smooth curve such that, R(2) =, R'(2) = , and R"(2) =…

A: Radius of curvature of teh function is given by formula, R = (1+y'(x)2)32y''(x)

Q: Find the tangent plane to the equation z = 2x – 3y + y at the point (1, 2, – 8). -

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Q: Find a vector parametrization of the circle of radius 5 in the xy-plane, centered at the origin,…

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Q: Find an equation of the tangent plane to the surface at the given point. f(x, y) = x² – 2xy + y²,…

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Q: Find the equation of the plane tangent to the surface of Z= x2+ Y2 at point P(125)

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Q: find the points on the surface + z² = 1 that have their tangent plane perpendicular to the vector…

A: For tangent plane of the surface f(x,y,z)=k at a given point normal vector to tangent plane  will be…

Q: Find the vector r' (to); then sketch the graph of r (t) in 2-space and draw the tangent vector r'…

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Q: Find the tangent plane to the equation z = x + 3y – 5y at the point ( – 5, – 5, 125) 2 Z =

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Q: Let f(r.y) = r'y- Find a unit vector tangent to the level curve f(r, y) = 3 at its point (1,2).

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Q: Consider the curve C in R3 whose parameterization is given by: eq. in image If is there a point on…

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Q: Find an equation of the tangent plane to the surface at the given point. h(x, y) = In Vx2+ y², (3,…

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Q: Find the equation of the plane tangent Z = (x+2)*- (y-1)²-5 at (2,3,3) to

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

A: given parametrically by x = x(t), y = y(t), and z = z(t) for a ≤ t ≤ b in the vector form.

Q: Find the tangent plane to the surface z = 9 – x* – y? at the point (1, -3, –1).

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Q: Find the tangent plane to the following surfaces at the point (1, 2, 3): a.) z = x + y     b.) z + 2…

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Q: Find the point(s) on the surface z = 3x2 + 2y2 − 3x + 4y − 5, at which the tangent plane is…

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Q: Let C be the curve of intersection of the surfaces with equations   x2 + y  =  5   and   24x + 4y…

A: Given: Let C be the curve of intersection of the surfaces with equations x2+y=5  and   24x + 4y + z…

Q: Find an equation of the tangent plane to the surface at the given point. h(x, y) = In Vx2 + y²,…

A: We have to find the equation of tangent plane

Q: ind an equation for the tangent plane of z = –2x²y – 3x² + y at the point (1, –1, –2).

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Q: Find an equation of the tangent plane to the surface at the given point. h(x, y) = In Vx2 + y², (12,…

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Q: Find the equation of the plane tangent to the surface z x+y at the point (2, 1, 3). x-1

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Q: Let r (t) = ti + e2t j + t³ k be a space curve. Which of the following vectors is perpendicular to…

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Q: Suppose r(t) is a vector function in R3 Show that if r(t)is perpendicular to the tangent vector…

A: Suppose that the position vector r→t be, r→t=ft,gt,ht…

Q: Let C be the portion of the parabola y = x², oriented from the starting point (0,0) towards the end…

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Q: Find the equation of the plane tangent to the surface f(x, y) = x In(x y) at the point (1, e).

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Q: Let r (t) = ti + e2t j + t3 k be a space curve. Which of the following vectors is perpendicular to…

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Q: Find the tangent plane to f(1, y) = ar² + 5y² + 71²y² +9x +5y at the point (3, 6). %3D

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Q: Find the equation of the tangent plane to the surface z = 3 e^(x−y) at the point (4, 4, 3)

A: We find the equation of the tangent plane to the surface z = 3ex-y at the point (4, 4, 3).

Q: Show that the vector-valued function r(t) = e−t cos ti + e−t sin tj + e−t k  lies on the cone z2 =…

A: Given vector valued function is r(t)=e-tcosti+e-tsintj+e-tk.

Q: Find a vector parametrization of the circle of radius 5 in the xy-plane, centered at the origin,…

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Q: Find the equation of the tangent plane to the surface z = +y? at the point (2, –2, 5)

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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.