Math 2.0

Geometry Definition

Normal vector

Builds on PlaneLineVectors are perpendicular if and only if their dot product is zero

Definition

Let HH be a line in ℝ2\mathbb{R}^2 or a plane in ℝ3\mathbb{R}^3. Let ⋅\cdot be the dot product. A normal vector of HH is a nonzero vector nn such that n⋅(x−y)=0n \cdot (x - y) = 0 for all points xx and yy of HH.

Remarks

If nn is a normal vector of HH, then the normal vectors of HH are exactly the vectors cnc n with c≠0c \neq 0. The plane {(x,y,z):ax+by+cz=d}\{(x, y, z) : a x + b y + c z = d\} with (a,b,c)≠(0,0,0)(a, b, c) \neq (0, 0, 0) has the normal vector (a,b,c)(a, b, c). The line {(x,y):ax+by=c}\{(x, y) : a x + b y = c\} with (a,b)≠(0,0)(a, b) \neq (0, 0) has the normal vector (a,b)(a, b). For the plane {p+su+tv:s,t∈ℝ}\{p + s u + t v : s, t \in \mathbb{R}\}, the cross product u×vu \times v is a normal vector. A unit normal vector is a normal vector of length 1.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Angle between two vectors, concept, no evidence levelAngle between two vec…conceptLength of a vector, concept, no evidence levelLength of a vectorconceptLine, concept, no evidence levelLineconceptPlane, concept, no evidence levelPlaneconceptGeometric formula for the dot product, statement, no evidence levelGeometric formula for…statementVectors are perpendicular if and only if their dot product is zero, statement, no evidence levelVectors are perpendic…statementNormal vector, definition, no evidence levelNormal vectordefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Angle between two vectors concept
  2. Length of a vector concept
  3. Line concept
  4. Plane concept
  5. Geometric formula for the dot product statement
  6. Vectors are perpendicular if and only if their dot product is zero statement
  7. Normal vector definition

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Normal_(geometry), Normal to planes and polygons
About this page
Tag0XYP2KKSPC
URIm2://0XYP2KKSPC
Kinddefinition
WritingW1 stated
MSC51N20
DefinesNormal vector
UsesPlane, Line, Vectors are perpendicular if and only if their dot product is zero

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