How would one find the equation for the normal line to a 3-dimensional equation at a given point?
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I have searched long and hard for this, but all that is discussed are tangent planes to a 3-D equation. I am aware of how to do derivates. If anyone could provide me with an equation, or a pointer in the right direction it would be much appreciated.
derivatives 3d
asked Jan 13 at 4:52
George CroftGeorge Croft
45
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add a comment |
$begingroup$
I have searched long and hard for this, but all that is discussed are tangent planes to a 3-D equation. I am aware of how to do derivates. If anyone could provide me with an equation, or a pointer in the right direction it would be much appreciated.
derivatives 3d
asked Jan 13 at 4:52
George CroftGeorge Croft
45
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$begingroup$
There is no unique tangent line. Given the tangent plane you can simply choose any line in that plane and it will be a tangent line at that point.
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– CyclotomicField
Jan 13 at 5:12
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I know it's a huge difference, but I have just edited my post to normal line, instead of tangent.
$endgroup$
– George Croft
Jan 13 at 22:32
add a comment |
$begingroup$
I have searched long and hard for this, but all that is discussed are tangent planes to a 3-D equation. I am aware of how to do derivates. If anyone could provide me with an equation, or a pointer in the right direction it would be much appreciated.
derivatives 3d
asked Jan 13 at 4:52
George CroftGeorge Croft
45
$endgroup$
I have searched long and hard for this, but all that is discussed are tangent planes to a 3-D equation. I am aware of how to do derivates. If anyone could provide me with an equation, or a pointer in the right direction it would be much appreciated.
derivatives 3d
derivatives 3d
asked Jan 13 at 4:52
George CroftGeorge Croft
45
asked Jan 13 at 4:52
George CroftGeorge Croft
45
edited Jan 13 at 22:32
George Croft
asked Jan 13 at 4:52
George CroftGeorge Croft
45
asked Jan 13 at 4:52
George CroftGeorge Croft
45
asked Jan 13 at 4:52
George CroftGeorge Croft
45
45
$begingroup$
There is no unique tangent line. Given the tangent plane you can simply choose any line in that plane and it will be a tangent line at that point.
$endgroup$
– CyclotomicField
Jan 13 at 5:12
$begingroup$
I know it's a huge difference, but I have just edited my post to normal line, instead of tangent.
$endgroup$
– George Croft
Jan 13 at 22:32
add a comment |
$begingroup$
There is no unique tangent line. Given the tangent plane you can simply choose any line in that plane and it will be a tangent line at that point.
$endgroup$
– CyclotomicField
Jan 13 at 5:12
$begingroup$
I know it's a huge difference, but I have just edited my post to normal line, instead of tangent.
$endgroup$
– George Croft
Jan 13 at 22:32
$begingroup$
There is no unique tangent line. Given the tangent plane you can simply choose any line in that plane and it will be a tangent line at that point.
$endgroup$
– CyclotomicField
Jan 13 at 5:12
$begingroup$
There is no unique tangent line. Given the tangent plane you can simply choose any line in that plane and it will be a tangent line at that point.
$endgroup$
– CyclotomicField
Jan 13 at 5:12
$begingroup$
I know it's a huge difference, but I have just edited my post to normal line, instead of tangent.
$endgroup$
– George Croft
Jan 13 at 22:32
$begingroup$
I know it's a huge difference, but I have just edited my post to normal line, instead of tangent.
$endgroup$
– George Croft
Jan 13 at 22:32
add a comment |
1 Answer
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A tangent line is characteristic of a curve - which isn't going to be defined by a single equation. Most commonly, a 3-D curve will be defined by parametric equations $(x(t),y(t),z(t))$, and the tangent is then a line in the direction of $(x'(t),y'(t),z'(t))$ - which, again, wouldn't be described as a single equation.
I recommend looking for a calculus textbook's treatment of vector calculus. Before the material with functions of multiple variables and multiple integrals, there should be a chapter dealing with curves and the like. (In Stewart, on a shelf near me, that's chapter 11)
[Added in edit]
For the normal line to a surface, we take the perpendicular to the tangent plane. The normal to a plane $a(x-x_0)+b(y-y_0)+c(z-z_0)$ at the point $(x_0,y_0,z_0)$ has vector equation $(x_0,y_0,z_0)+t(a,b,c)$ - the vector difference of two points on the line is a multiple of the vector of coefficients of the plane's equation. Coordinate-wise, that's $(x_0+ta,y_0+tb,z_0+tc)$. Of course, $t$ is a dummy variable for the parameter here; any convenient name that doesn't conflict with existing variable names can be chosen.
answered Jan 13 at 5:30
jmerryjmerry
5,727616
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1 Answer
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$begingroup$
A tangent line is characteristic of a curve - which isn't going to be defined by a single equation. Most commonly, a 3-D curve will be defined by parametric equations $(x(t),y(t),z(t))$, and the tangent is then a line in the direction of $(x'(t),y'(t),z'(t))$ - which, again, wouldn't be described as a single equation.
I recommend looking for a calculus textbook's treatment of vector calculus. Before the material with functions of multiple variables and multiple integrals, there should be a chapter dealing with curves and the like. (In Stewart, on a shelf near me, that's chapter 11)
[Added in edit]
For the normal line to a surface, we take the perpendicular to the tangent plane. The normal to a plane $a(x-x_0)+b(y-y_0)+c(z-z_0)$ at the point $(x_0,y_0,z_0)$ has vector equation $(x_0,y_0,z_0)+t(a,b,c)$ - the vector difference of two points on the line is a multiple of the vector of coefficients of the plane's equation. Coordinate-wise, that's $(x_0+ta,y_0+tb,z_0+tc)$. Of course, $t$ is a dummy variable for the parameter here; any convenient name that doesn't conflict with existing variable names can be chosen.
answered Jan 13 at 5:30
jmerryjmerry
5,727616
$endgroup$
add a comment |
$begingroup$
A tangent line is characteristic of a curve - which isn't going to be defined by a single equation. Most commonly, a 3-D curve will be defined by parametric equations $(x(t),y(t),z(t))$, and the tangent is then a line in the direction of $(x'(t),y'(t),z'(t))$ - which, again, wouldn't be described as a single equation.
I recommend looking for a calculus textbook's treatment of vector calculus. Before the material with functions of multiple variables and multiple integrals, there should be a chapter dealing with curves and the like. (In Stewart, on a shelf near me, that's chapter 11)
[Added in edit]
For the normal line to a surface, we take the perpendicular to the tangent plane. The normal to a plane $a(x-x_0)+b(y-y_0)+c(z-z_0)$ at the point $(x_0,y_0,z_0)$ has vector equation $(x_0,y_0,z_0)+t(a,b,c)$ - the vector difference of two points on the line is a multiple of the vector of coefficients of the plane's equation. Coordinate-wise, that's $(x_0+ta,y_0+tb,z_0+tc)$. Of course, $t$ is a dummy variable for the parameter here; any convenient name that doesn't conflict with existing variable names can be chosen.
answered Jan 13 at 5:30
jmerryjmerry
5,727616
$endgroup$
add a comment |
$begingroup$
A tangent line is characteristic of a curve - which isn't going to be defined by a single equation. Most commonly, a 3-D curve will be defined by parametric equations $(x(t),y(t),z(t))$, and the tangent is then a line in the direction of $(x'(t),y'(t),z'(t))$ - which, again, wouldn't be described as a single equation.
I recommend looking for a calculus textbook's treatment of vector calculus. Before the material with functions of multiple variables and multiple integrals, there should be a chapter dealing with curves and the like. (In Stewart, on a shelf near me, that's chapter 11)
[Added in edit]
For the normal line to a surface, we take the perpendicular to the tangent plane. The normal to a plane $a(x-x_0)+b(y-y_0)+c(z-z_0)$ at the point $(x_0,y_0,z_0)$ has vector equation $(x_0,y_0,z_0)+t(a,b,c)$ - the vector difference of two points on the line is a multiple of the vector of coefficients of the plane's equation. Coordinate-wise, that's $(x_0+ta,y_0+tb,z_0+tc)$. Of course, $t$ is a dummy variable for the parameter here; any convenient name that doesn't conflict with existing variable names can be chosen.
answered Jan 13 at 5:30
jmerryjmerry
5,727616
$endgroup$
A tangent line is characteristic of a curve - which isn't going to be defined by a single equation. Most commonly, a 3-D curve will be defined by parametric equations $(x(t),y(t),z(t))$, and the tangent is then a line in the direction of $(x'(t),y'(t),z'(t))$ - which, again, wouldn't be described as a single equation.
I recommend looking for a calculus textbook's treatment of vector calculus. Before the material with functions of multiple variables and multiple integrals, there should be a chapter dealing with curves and the like. (In Stewart, on a shelf near me, that's chapter 11)
[Added in edit]
For the normal line to a surface, we take the perpendicular to the tangent plane. The normal to a plane $a(x-x_0)+b(y-y_0)+c(z-z_0)$ at the point $(x_0,y_0,z_0)$ has vector equation $(x_0,y_0,z_0)+t(a,b,c)$ - the vector difference of two points on the line is a multiple of the vector of coefficients of the plane's equation. Coordinate-wise, that's $(x_0+ta,y_0+tb,z_0+tc)$. Of course, $t$ is a dummy variable for the parameter here; any convenient name that doesn't conflict with existing variable names can be chosen.
answered Jan 13 at 5:30
jmerryjmerry
5,727616
edited Jan 13 at 22:50
answered Jan 13 at 5:30
jmerryjmerry
5,727616
answered Jan 13 at 5:30
jmerryjmerry
5,727616
answered Jan 13 at 5:30
jmerryjmerry
5,727616
5,727616
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add a comment |
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$begingroup$
There is no unique tangent line. Given the tangent plane you can simply choose any line in that plane and it will be a tangent line at that point.
$endgroup$
– CyclotomicField
Jan 13 at 5:12
$begingroup$
I know it's a huge difference, but I have just edited my post to normal line, instead of tangent.
$endgroup$
– George Croft
Jan 13 at 22:32