# What is Normal line: Definition and 28 Discussions

In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point.
A normal vector may have length one (a unit vector) or its length may represent the curvature of the object (a curvature vector); its algebraic sign may indicate sides (interior or exterior).
In three dimensions, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the tangent plane of the surface at P. The word "normal" is also used as an adjective: a line normal to a plane, the normal component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality (right angles).
The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point P is the set of vectors which are orthogonal to the tangent space at P.
Normal vectors are of special interest in the case of smooth curves and smooth surfaces.
The normal is often used in 3D computer graphics (notice the singular, as only one normal will be defined) to determine a surface's orientation toward a light source for flat shading, or the orientation of each of the surface's corners (vertices) to mimic a curved surface with Phong shading.

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1. ### Calculating the equations for the tangent/normal lines

9. When I do this problem I know my slope is -3 because f'(2)=-3. I then went and substituted and got y+5=-3(x-2) which simplified to y=-3x+1 10. I get lost here because the tangent slope would be 0, which would give me the equation y=-2. The normal means perpendicular and the perpendicular...
2. ### Unit Normal to a level surface

Homework Statement Given $$\phi = x^{2} +y^{2}-z^{2}-1$$ Calculate the unit normal to level surface φ = 0 at the point r = (0,1,0) Homework Equations $$\hat{\mathbf n} = \frac{∇\phi}{|\phi|}$$ $$z = \sqrt{x^{2}+y^{2} -1}$$  \mathbf n = (1,0,(\frac{\partial z}{\partial x})_{P})...

Hi All, This question is about vector calculus, gradient, directional derivative and normal line. If the gradient is the direction of the steepest ascent: >> gradient(x, y) = [ derivative_f_x(x, y), derivative_f_y(x, y) ] Then it really confuse me as when calculating the normal line...
4. ### Logic behind normal line in expressing plane

why do we consider normal line in expressing a plane,say in ##R^3## ,of the form ## ax + by + cz = d ##? What is the logic behind this normal line selection? Plz provide intuitive explanations.Thanks
5. ### Equation of normal line and tangent plane

Homework Statement A cup is represented by the surface -(z-1)2 + x2 + y2 = 1 and it is on a table represented by the plane z=0 a) find the angle at which the cup intersects the table b) find the equation of the normal line to the cup at the point (0, √2 , 2) c) find the equation the tangent...
6. ### Finding point on paraboloid surface given normal line point

Homework Statement [/B] Find the coordinates of the point P on the surface of the paraboloid z=6x2+6y2-(35/6) where the normal line to the surface passes through the point (25/6, (25√22)/6, -4). Note that a graphing calculator may be used to solve the resulting cubic equation. Homework...
7. ### How do I find the equation of the normal line for a given function and point?

Homework Statement Find the equation of the normal line of function f(x,y) = x^2+y^2-1 through points (2,1,4) Homework Equations (x-xo)/f_x = (y-yo)/f_y = (z-zo)/f_z The Attempt at a Solution so... f_x = 2x = 4 f_y = 2y = 2 f_z = ? I don't know how to solve for f_z... here...
8. ### What is the slope of the normal line at x=3 for the inverse of f(x)=5x2+3e2x?

Homework Statement The function f(x)=5x2+3e2x is invertible. Give the slope of the normal line to the graph of f-1 at x=3 Homework Equations (f-1)'(x)=1/f'(f-1(x)) The slope of the normal line × the slope of the tangent =-1 So basically I just need to find -f'(f-1(x)) at x=3 The Attempt at a...
9. ### Finding Equations of Normal Lines Parallel to a Given Line

Homework Statement Find the equation of each of the normal lines to the curve y = x^3 - 4x that is parallel to the line x + 8y - 8 = 0. Homework Equations Differentiation, y - y1 = m(x - x1) The Attempt at a Solution Well, clearly I start by differentiating y = x^3 - 4x to gey...
10. ### MHB Find Equation of Normal Line to y=x^2-5x+4 Parallel to x-3y=5

Hello, i know how to calculate normal line on point but i don't know how to do this problem. "find an equation of the normal line o the parabola y=x^2-5x+4 tha is parallel to the line x-3y=5."
11. ### Where does the normal line intersect the second time?

Where does the normal line to the ellipse x2-xy+y2=3 at the point (-1,1) intersect the ellipse a second time?So I took the derivative of the equation to get: y'=(2x-y)/(x-2y) Then I put (-1,1) into the equation, to get a slope of 1. So, for the normal line I got a slope of -1, which has the...
12. ### Points that a curve's normal line intersects

Homework Statement The line that is normal to the curve x2+2xy-3y2=0 at (5,5) intersects the curve at what other point? 2. The attempt at a solution I differentiated the equation, found the slope of the curve at that point, and I then found the equations for the tangent line and normal...
13. ### Finding the tangent plane and normal line

D is a set of all points (x,y) in R2 distinct from (0,0). I have the funtion f: D --> R which is defined by: f(x,y) = (2xy)/(x2+y2 Im asked to find the equations (in plural) of the tangelnt plane and the normal line to the graph of the function at (1,2) My attempt: I use the tangent plane...
14. ### Principal normal line to helix is normal to z axis

Homework Statement DoCarmo Section 1.5 problem 1 part d. Show that the lines containing n(s) and passing through a(s) [a is the curve, and n(s) is the unit normal vector] meet the z axis under a constant angle of pi/2. Helix: a(s) = (a cos(s/c_, a sin(s/c), b*s/c), so I computed n(s) =...
15. ### Intersection of paraboloid and normal line

Homework Statement Where does the normal line to the paraboloid z=x^2+y^2 at the point (1,1,2) intersect the paraboloid a second time? Homework Equations The Attempt at a Solution I found the normal line to be 0=2x+2y-1, but I'm not sure what to do next.
16. ### Finding equation of tangent plane & normal line to a given surface

Homework Statement Find equations of the tangent plant and the normal line to x-z=4arctan(yz) at (1+∏, 1, 1) Homework Equations The Attempt at a Solution I took the partials and got fx=0 fy=0 fz=(-4y)/((yz)2+1) so for the plane i got -2z-2=0 and for the normal line I got .5z-2=0...
17. ### Proving Every Normal Line to a Sphere Passes Through Center

Homework Statement Prove that every normal line to a sphere passes through the centre of the sphere.Homework Equations Equation of the sphere with it's center at O(a,b,c): (x-a)²+(y-b) ²+(z-c)²=r² or (x-a)²+(y-b) ²+(z-c)²-r² = 0 = F(x,y,z)Parametric equations of the normal line at...
18. ### Equation of a Tangent plane an the normal line to a given point

Homework Statement xy +yz + zx = 3 (1,1,1) Homework Equations equation of tangent plane is z-z0 = fx(x0,y0)(x-x0) +fy(x0,y0)(y-y0) The Attempt at a Solution Right, I've been a few of these exercises, however, this is the first one I've seen that equals a number and not "z". So...
19. ### Tangent Line and Normal Line

Homework Statement Find the lines that are tangent and normal to the curve x^2 + xy - y^2=1 at (2,3) Homework Equations Umm.. first derivative and second derivative? The Attempt at a Solution For the tangent line i found the first derivative and used the implicit differentiation(?)...
20. ### Finding the Normal Line for a Curve at a Given Point

find the equation of the normal line to the curve y = -3x^2 at the point (-1,-3). you have to use derivatives. i am not asking for the equation of the tangent line i know that is 6x + 3. please show all the steps. the answer is y = -1/6x - 19/6
21. ### Parallel Normal Lines between y = sqrt(x - 1) and y = 1 - 2x

Homework Statement Find the points where the normal line to the curve y = sqrt(x - 1) is parallel to the line y = 1 - 2x. The Attempt at a Solution m=-2 y' = 1/2sqrt(x-1) = m = -2 -4sqrt(x-1)=1 sqrt(x-1) = - 1/4 x= 1/16 + 1 x= 17/16 so find y y = sqrt((17/16)-1)) (17/16, 1/4) this is...
22. ### Parabola, tangebt line and Normal Line intersect

Homework Statement Where does the normal line to the parabola y= x - x^2 at the point (1,0) intersect the parabola a second time? Illustrate with a sketch Homework Equations y= x - x^2 The Attempt at a Solution y' = 1-2x slope of tangent = -1 (after substituting value of x in...
23. ### Finding the equation of the normal line at

Homework Statement Find the equation, in standard form, of the normal line drawn to the graph of y=3x4 - 1/x at x = 1 Homework Equations The Attempt at a Solution -I figured out the slope of the tangent to be 12x3 + 1, then at x = 1 the slope is 13 -SO the slope of the...
24. ### Find Normal Line to Parabola y=x^2-5x+4

Homework Statement Find an equation of the normal line to the parabola y = x^2 - 5x + 4 that is parallel to the line x - 3y = 5. Homework Equations ? All that is relevant is mentioned below AFAIK. The Attempt at a Solution Here is my attempt: http://screencast.com/t/oXxI6JPHv My...
25. ### Normal line to a tangent plane

Homework Statement I have two problems that don't seem to match. The task is to find the parametric equation for the normal line to the tangent plane. I'm seeing that the vector value of "z" is (-1) with one formula, but not in the other. These two seem to be the same, but the first was solved...
26. ### Equation of Normal Line: 6x + y + 9 = 0

Homework Statement Find the equation of the normal line to the curve y(sqrd) - x(sqrd)y + 3 = 0 at the point (-2,3). (standard form) Homework Equations y - y = m(x - x) The Attempt at a Solution dy __ = -6 dx y - 3 = -6(x + 2) 6x + y +9 = 0
27. ### What is the difference between a normal line and a tangent line?

What is the difference between a normal line and a tangent line?
28. ### Where Does the Normal Line Intersect the Parabola Again?

Here is what I found, look at the end to see what my question is: At what point does the normal line to the graph of y= -4 - 3x + 4x^2 at (1, -3 ) intersect the parabola a second time? So far I have tried to find the derivative of the parabola which is: 8x-3 (this corresponds to the...