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Definition and Cartesian representation

Definition of a vector

A vector in the plane or in space is characterised by its magnitude, or length, and its direction, which is an oriented line (sometimes represented as a line with an arrow).

In other words, a vector connects two points in space by a straight line. Unlike a line segment, direction matters.

Generally, we identify a vector with the end point obtained by sliding the vector to bring the initial point to the origin.

Vectors have direction: $\uvec$ and $\vvec$ have the same direction, and they both have the opposite direction to $\wvec$

Position and displacement

A position vector represents a point in space: it starts at the origin $O$ and ends at a given point $A$. It is denoted $\overrightarrow{OA}$.

A displacement vector starts at an initial point $A$ and ends at a terminal point $B$. It is the difference between two position vectors, and represents the move in a straight line from $A$ to $B$. It is denoted $\overrightarrow{AB}$.

The displacement from $A$ to $B$ is the opposite of the displacement from $B$ to $A$, and so we have: $$\overrightarrow{AB} = -\overrightarrow{BA}$$ The length and the support line of the two vectors is the same; but the orientations are opposite.

Position ($\overrightarrow{OA}$) and displacement ($\overrightarrow{AB}$) vectors

Cartesian representation of a vector

In the Cartesian coordinate system, a vector is represented by the coordinates of its endpoint (when the origin is the initial point).

This can either be two dimensional space, or three dimensional space. In two dimensional space we use $\ivec$ and $\jvec$, and in three dimensional space we also use $\kvec$ for the extra dimension.

$$\uvec = x\ivec + y\jvec = (x,y),\qquad \vvec = x\ivec+y\jvec+z\kvec = (x,y,z).$$

The length of a vector is given by the formula $$\vert \uvec\vert = \sqrt{x^2+y^2},\qquad \vert \vvec\vert = \sqrt{x^2+y^2+z^2}.$$

Vectors in a coordinate system