Linear Algebra - Plane

> Linear Algebra

1 - About

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.

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3 - Type

3.1 - 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}

3.1.1 - 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}

is

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

3.2 - Translation

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
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3.2.1 - 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: c + nu

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

3.2.2 - Affine hull

3.2.3 - Equation

The solution set of an linear equation:

  • ax + by + cz = d
  • In vector terms:{[x, y, z] : [a, b, c] · [x, y, z] = d}
linear_algebra/plane.txt · Last modified: 2015/06/28 14:16 by gerardnico