Definition:Projection (Mapping Theory)

Let $$S$$ and $$T$$ be sets.

First Projection
The first projection on $$S \times T$$ is the mapping $$pr_1: S \times T \to S$$ defined by:

$$\forall \left({x, y}\right) \in S \times T: pr_1 \left({x, y}\right) = x$$

This is sometimes referred to as the projection on the first co-ordinate.

Second Projection
The second projection on $$S \times T$$ is the mapping $$pr_2: S \times T \to T$$ defined by:

$$\forall \left({x, y}\right) \in S \times T: pr_2 \left({x, y}\right) = y$$

This is sometimes referred to as the projection on the second co-ordinate.

Generalized Definition
Let $$S_1, S_2, \ldots, S_j, \ldots, S_n$$ be sets.

For each $$j \in \left[{1 \,. \, . \, n}\right]$$, the $$j$$th projection on $$S = \prod_{i=1}^n S_i$$ is the mapping $$pr_j: S \to S_j$$ defined by:

$$pr_j \left({s_1, s_2, \ldots, s_j, \ldots, s_n}\right) = s_j$$

for all $$\left({s_1, \ldots, s_n}\right) \in S$$.

This is sometimes referred to as the projection on the $$j$$th co-ordinate.

Coordinate Geometry
Let $$M$$ and $$N$$ be distinct lines through the plane through the origin.

The projection on $$M$$ along $$N$$ is the mapping $$p_{M, N}$$ such that $$\forall x \in \mathbb{R}^2: p_{M, N} \left({x}\right) =$$ the intersection of $$M$$ with the line through $$x$$ parallel to $$N$$.