Points in Geometry

Definition: Point

A point in the Euclidean space ${\mathbb{R}}^n$ is a real column vector.

INTUITION

We usually think of points as representing locations in space.

NOTATION

• Points are usually denoted with capital Latin letters without an arrow: $P,Q,K,L,$ etc.
• Instead of writing $P={\left[\begin{array}{ccc}{p}_1& \cdots & p_n\end{array}\right]}^{\mathsf{T}}$, we often write points as $n$-tuples $P=(p_1,\cdots ,p_n)$.

Definition: Vector

A vector in $n$-dimensional Euclidean space is a real column vector ${\mathbb{R}}^n$.

INTUITION

In geometry, we tend to think of vectors as arrows which point from the origin to some location in space, i.e. we think of vectors as having a length and a direction. This way of thinking is especially useful in $3$ and $2$ dimensions, since it can be easily visualised.

Notation: Vector between Points

If $P$ and $Q$ are two points, then we use $\overrightarrow{PQ}$ to denote the vector $Q-P$.

Intuition: Points vs Vectors

[PoinPointsvectors in Euclidean geometry are the same thing - they are real column vectors in the Euclidean space ${\mathbb{R}}^n$. We simply use different terminology and / or notation when we want to emphasise different aspects of those real column vectors.

• Point terminology is used when we are more interested in the location which the real column vector represents, i.e. we want to focus our attention on its components.
• Vector terminology is used when we are more interested in the vectors’s length