(a) The surface S is defined by x + y² + z³ = 3. Find a vector normal to S at the (1, 1, 1). Hence, find the equation of the plane = point A with position vector a tangent to S at A.

Q: Find parametric equations for the line through the point P(3, 6, 4) that is parallel to the plane x…

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Q: Let P be the point (1, 1, 4). Find the parametric equations of the line through P and parallel to…

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Q: Find an equation of the tangent plane to the surface z = -2z² – 2y?- lz 2y+ 3 at the point (5, 1,…

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Q: he vector v = , is tangent to the surface x2 + 2y3 - 3z2 = 3 at the point (2, 1, 1).  Find a.

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Q: Consider the surface of equation z = x2 + y² . What is the equation for the tangent line to S at the…

A: We need to find equation of tangent line.

Q: The vector parametric equation for the line through the points (-3, 5, −4) and (5, −5, 0) is L(t) =

A: The given points are -3, 5, -4 and 5, -5, 0. We have to find a parametric equation.

Q: Find the distance between point A(-1,4,6) and the line of parametril equations x = 2-3t, y = -4+t, z…

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Q: The surface z = x² - y² and the surface ryz = -30 intersect along a curve in R³. Find a vector…

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Q: Sketch the parametric curve of r(t) =.Then, find and draw the tangent vector r'(0)

A: 2cos t, 2sin t,3To Find:a. Sketch the parametric curve.b. Find and draw the tangent vector r'(0).

Q: Find an equation for the plane tangent to the level surface x² + y? +z = 4 at the point Po(1,1, 2).…

A: Solving

Q: Find the distance between the point (0,0, 8) and the plane given by 2x – y + 2z – 1 = 0.

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Q: By vector methods show that the line x = y = (1/3)(z + 2) is parallel to the plane 2x - 8y +2z = 5..

A: Any arbitrary point P0 on the line can be determined to be as:

Q: Consider the curve F(t) that represents the intersection of the surfaces x + y? -1=0 and y= 2x a.…

A: Given: The curve r(t) represents the intersection of the surfaces x2+y2-1=0 and y=z/2x.