What is Perpendicular vector: Definition and 16 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. Abdu Ewais

    I Problem with gravitation field perpendicular vector.

    since it is known that ##\vec{A_\perp} = -{mG \over R^2}## why did the professor write it as ##\vec{A_\perp} = {- R G \rho \over 3}## for perfect sphere with perfect mass distribution ? Shouldn't it be ##\vec{A_\perp} = -{4 \over 3} \pi R G \rho##? I need help thanks.
  2. M

    B How to calculate a perpendicular line?

    Hi, i get the math that is involved but if I have only the x,y coordinates for 2 points to connect and if i want to know what will be the perpendicular line to the line connecting these two points, how can I do that?
  3. Ventrella

    B Complex products: perpendicular vectors and rotation effects

    My question is perhaps as much about the philosophy of math as it is about the specific tools of math: is perpendicularity and rotation integral and fundamental to the concept of multiplication - in all number spaces? As I understand it, the product of complex numbers x = (a, ib) and y = (c...
  4. henry wang

    I Proving grad operator yields perpendicular vector to contour

    I saw a link on MIT open courseware proving grad operator yields perpendicular vector to contour, but I can't make sense of how dg/dt=0. Can someone explain to me please...
  5. AllenFaust

    Finding perpendicular vector in a skewed coordinate system

    Homework Statement I have an a-b coordinate system which is skewed with an angle = 60 deg. I also have a particle position defined by vector V1 (a1, b1, 0) which follows the coordinate system. The problem I have is that I need to get V2 (a2, b1, 0) which is perpendicular to V1. Homework...
  6. kostoglotov

    Beginner: understanding an answer

    Doing MIT OCW 18.06 using Gilbert Strang Intro to Linear Algebra. Ch 1.2 The vectors that are perpendicular to <1,1,1> and <1,2,3> lie on a ___________. I would have said "plane". I've worked with vectors in calculus, and if you take the cross product of those two vectors you get a vector...
  7. C

    Finding the Perpendicular Vector Equation

    Homework Statement Find the equation of the line, p, that is perpendicular to r=(1,1,1)t + (3,5,2) and passes through (3,3,1) Homework Equations dot product =0 The Attempt at a Solution I understand that the direction vector r and p, when dot producted must =0 but I've forgotten...
  8. PhizKid

    How to Find a Unit Vector Perpendicular to Another in the XY Plane?

    Homework Statement Find a unit vector in the xy plane which is perpendicular to A = (3,5,1). Homework Equations A_x{B_{x}} + A_y{B_{y}} + A_z{B_{z}} = \textbf{A} \cdot \textbf{B}\\\hat{\textbf{A}} = \frac{\textbf{A}}{|\textbf{A}|} The Attempt at a Solution In order to be perpendicular...
  9. S

    Perpendicular vector in 6th dimensional space

    I am working on a software for analysis of brain connections using MRI. Please suggest the simplest way to find a vector which is: perpendicular to a unit vector that is positioned in the coordinate center, it should be in the 2D plane containing the given vector and a given point on...
  10. A

    Perpendicular vector using dot not cross product.

    Homework Statement I have 2 (3d)vectors A and B and I want to find a vector C perpendicular to both of them. A = 3i-2j+4k B = -2i+5j-2k C = Cx+Cy+Cz Homework Equations So we know A dot C = 3Cx-2Cy+4Cz and B dot C = -2Cx+5Cy-2Cz The Attempt at a Solution
  11. N

    Perpendicular Vector with given magnitude

    Homework Statement Given a vector A = 8i − 5j, find the vectors in the xy plane that are perpendicular to A and have a magnitude of 14. The Attempt at a Solution I can find the vector perpendicular to A quite easily. I'm having trouble finding the perpendicular line when it has the...
  12. M

    Find Value of \alpha for Perpendicular Vector to Plane

    Homework Statement I'm given a vector (x,y,\alpha) (cant state exact number cause of plagarism). I have to find the value of \alpha which make the vector perpendicular to a plane. I also have 2 other vectors that are parallel to the plane. Homework Equations dot product. The...
  13. R

    Finding the Unique Perpendicular Vector at P=(3,4,5)

    Is it correct? unique perpendicular vector Homework Statement Find the unique vector which is perpendicular on z= \sqrt{x^2+y^2} at P=(3,4,5) Homework Equations \frac{\nabla F}{\left| \nabla F\left|} The Attempt at a Solution The answer is \frac{\nabla F}{\left| \nabla...
  14. E

    Find Basis of Perpendicular Vectors for v1 & v2

    Homework Statement how do I find the basis for all vectors that are perpendicular to both v1 = (1,1,0,0) and v2 = (1,0,1,1) If it were 3-D I could find a normal vector via cross product, but in n-dimensions... what do I do? My one thought is to arrange and solve linear homogeneous system...
  15. E

    Perpendicular Vector at Point P on Surface x^3 + xy^3 - z^2 = -4

    The question reads: "Consider the surface given by the equation x^3 + xy^3 - z^2 = -4 The point p = (1,2,3) lies on this surface. Give a vector that is perpendicular to the surface at p?" I'm not too confident about this question although there is a theorem in my notes saying: if p...
  16. S

    Find Perpendicular Vector w/ Dot Product

    If you had two vectors, and you wanted to find a vector perpendicular to those useing th dot product, what would be needed to be done. I alreaded Doted A *dot* B and have a vector. I'm just stuck on the last part