Linear Algebra - Plane

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A plane is a two dimensional vector space.

A plane has a dimension of two because two coordinates are needed to specify a point on it.


Containing the origin

  • Two-dimensional: All points in the plane: Span {[1, 2], [3, 4]}
  • Span of two 3-vectors {[1, 0, 1.65], [0, 1, 1]} is a plane in three dimensions.
# A more familiar way to specify a plane
{(x, y, z) : ax + by + cz = 0}
# Using dot-product, the above equation becomes a set of vectors 
# satisfying a linear equation with right-hand side zero
{[x, y, z] : [a, b, c] * [x, y, z] = 0}

Plane Intersection

The intersection of the two following plane:

  • {[x, y, z] : [4,-1, 1] · [x, y, z] = 0}
  • {[x, y, z] : [0, 1, 1] · [x, y, z] = 0}


  • {[x, y, z] : [4,-1, 1] · [x, y, z] = 0, [0, 1, 1] · [x, y, z] = 0}

Vector Plan Intersection


The translation of a plane translate it in a way that it doesn't contain the origin.

You can express such plane as

  • a vector addition
  • an affine hull
  • a solution set of an equation

Vector Addition

Vector addition is used to defined a set of points forming an plane that not necessarily go through the origin.

You translate the plane by adding a vector c [0.5, 1] to every point in the plane.

delim{lbrace}{c + v : v in nu}{rbrace}

abbreviated: <math>c + nu</math>

Mathematics Plane Translation

The result is a plane through c instead of through origin.

Affine hull


The solution set of an linear equation:

  • ax + by + cz = d
  • In vector terms:{[x, y, z] : [a, b, c] · [x, y, z] = d}

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