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Find a unit normal vector to the surface x^2 y^2-2z 3=0 at 1,2,-1.?
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Find a unit normal vector to the surface x^2 y^2-2z 3=0 at 1,2,-1.?
Unit Normal Vector to the Surface
To find a unit normal vector to the surface x^2 y^2 - 2z - 3 = 0 at the point (1, 2, -1), we can follow these steps:
1. Calculate the Gradient of the Surface
The gradient of the surface is given by the vector:
grad(f) =
For the given surface x^2 y^2 - 2z - 3 = 0, the gradient is:
grad(f) = <2xy^2, 2x^2y,="" -2="">
2. Calculate the Gradient at the Given Point
Substitute the coordinates of the point (1, 2, -1) into the gradient vector:
grad(f) at (1, 2, -1) = <2*(1)*(2)^2, 2*(1)^2*(2),="" -2=""> = <8, 4,="" -2="">
3. Normalize the Gradient Vector
To find the unit normal vector, we need to normalize the gradient vector by dividing each component by the magnitude of the vector:
||grad(f) at (1, 2, -1)|| = sqrt(8^2 + 4^2 + (-2)^2) = sqrt(84) = 2sqrt(21)
Normalized gradient vector = <8 2sqrt(21)),="" 4/(2sqrt(21)),="" -2/(2sqrt(21))="">
= <4 qrt(21),="" 2/sqrt(21),="" -1/sqrt(21)="">
4. Unit Normal Vector
Therefore, the unit normal vector to the surface x^2 y^2 - 2z - 3 = 0 at the point (1, 2, -1) is <4 qrt(21),="" 2/sqrt(21),="" -1/sqrt(21)="">.
To find a unit normal vector to the surface x^2 y^2 - 2z - 3 = 0 at the point (1, 2, -1), we can follow these steps:
1. Calculate the Gradient of the Surface
The gradient of the surface is given by the vector:
grad(f) =
For the given surface x^2 y^2 - 2z - 3 = 0, the gradient is:
grad(f) = <2xy^2, 2x^2y,="" -2="">
2. Calculate the Gradient at the Given Point
Substitute the coordinates of the point (1, 2, -1) into the gradient vector:
grad(f) at (1, 2, -1) = <2*(1)*(2)^2, 2*(1)^2*(2),="" -2=""> = <8, 4,="" -2="">
3. Normalize the Gradient Vector
To find the unit normal vector, we need to normalize the gradient vector by dividing each component by the magnitude of the vector:
||grad(f) at (1, 2, -1)|| = sqrt(8^2 + 4^2 + (-2)^2) = sqrt(84) = 2sqrt(21)
Normalized gradient vector = <8 2sqrt(21)),="" 4/(2sqrt(21)),="" -2/(2sqrt(21))="">
= <4 qrt(21),="" 2/sqrt(21),="" -1/sqrt(21)="">
4. Unit Normal Vector
Therefore, the unit normal vector to the surface x^2 y^2 - 2z - 3 = 0 at the point (1, 2, -1) is <4 qrt(21),="" 2/sqrt(21),="" -1/sqrt(21)="">.
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Find a unit normal vector to the surface x^2 y^2-2z 3=0 at 1,2,-1.?
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Find a unit normal vector to the surface x^2 y^2-2z 3=0 at 1,2,-1.? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Find a unit normal vector to the surface x^2 y^2-2z 3=0 at 1,2,-1.? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find a unit normal vector to the surface x^2 y^2-2z 3=0 at 1,2,-1.?.
Find a unit normal vector to the surface x^2 y^2-2z 3=0 at 1,2,-1.? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Find a unit normal vector to the surface x^2 y^2-2z 3=0 at 1,2,-1.? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Find a unit normal vector to the surface x^2 y^2-2z 3=0 at 1,2,-1.?.
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