Vectors and Coordinate System

Cartesian Coordinate System

To specify the position of an object, such as an aircraft in flight, we need a frame of reference or coordinate system. The most commonly used system is the Cartesian Coordinate System, developed by René Descartes.

The Cartesian Coordinate System uses perpendicular axes to determine the location of a point.

• In two dimensions (2D), we use the $x$ axis (horizontal) and $y$ axis (vertical). A point's position is expressed as an ordered pair $(x, y)$, for example, point $P(3, 4)$.

  This means that point $P$ is located $3$ units to the right (positive $x$ axis) and $4$ units upward (positive $y$ axis) from the origin $O(0, 0)$.

  Position Vector $P(3, 4)$

  Origin $O(0, 0)$ to point $P(3, 4)$

• In three dimensions (3D), we add a $z$ axis that is perpendicular to the $xy$ plane. A point's position is expressed as $(x, y, z)$, used for determining the position of objects like aircraft in space.

  We can think of $x$ as length, $y$ as width, and $z$ as height.

  Position Vector $P(3, 4, 5)$

  Origin $O(0, 0, 0)$ to point $P(3, 4, 5)$

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Difference Between Vectors and Coordinates

Unlike coordinates which represent the position of a point, a vector is a quantity that has both magnitude (length) and direction. Imagine a fly traveling from point $P(3, 4)$ to point $Q(-2, -8)$. The fly's displacement can be represented as a vector.

• Position Vector: A vector starting from the origin $O(0, 0)$ to a point $P(x, y)$. The position vector of $P$ is written as $\overrightarrow{OP}$.

  Example: $\overrightarrow{OP} = (3, 4)$.

• Displacement Vector (or Free Vector): A vector connecting any two points, for example, from point $P(x_P, y_P)$ to point $Q(x_Q, y_Q)$. This vector is written as $\overrightarrow{PQ}$. It doesn't have to start from the origin.

  Example: displacement vectors from $P(3, 4)$ to $Q(-2, -8)$ or from $Q(-2, -8)$ to $R(1, 3)$.

  Displacement Vectors

  Displacement vectors from $P$ to $Q$ or from $Q$ to $R$

Vector Notation

Vectors can be written in several ways:

• Row Vector: Using regular parentheses.

  Example: $\overrightarrow{OP} = (3, 4)$.

• Column Vector: Using square brackets or large parentheses.

  Example: $\overrightarrow{OP} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$.

• Unit Vector Notation (for 3D): Using a combination of unit vectors $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ which are aligned with the $x$, $y$, and $z$ axes.

  Example: a vector $\overrightarrow{AB}$ can be written as $\mathbf{i} - 3\mathbf{j} - 4\mathbf{k}$, meaning vector $\overrightarrow{AB}$ has a component of $1$ in the $x$ direction, $-3$ in the $y$ direction, and $-4$ in the $z$ direction.

Why Are Coordinate Systems Important?

Imagine a map application on your phone. This application relies on coordinate systems to:

1. Determine your current location.
2. Find your destination.
3. Calculate the shortest or fastest route.
4. Provide direction instructions (turn left, turn right, go straight).

Without coordinate systems, it would be impossible for applications to accurately understand position and direction. Every movement, such as an aircraft changing altitude or direction, needs to be reported and processed using coordinate systems and vector concepts.