# Linear Algebra - Plane

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.

## 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

• an affine hull
• a solution set of an equation

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.

abbreviated: $c + nu$

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

#### 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}

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