Two-Dimensional Vector

Two-Dimensional Vector Concept

In the Cartesian coordinate system, each point on a plane can be represented by a pair of numbers $(x, y)$ , where $x$ is the horizontal position and $y$ is the vertical position. The origin point is $O(0, 0)$ .

If we draw a straight line from the origin $O$ to another point, for example $Q(x, y)$ , we get a vector. This vector is often written as $\overrightarrow{OQ}$ . A vector has both magnitude (line length) and direction (indicated by the arrow).

To simplify, we use unit vectors. Unit vectors have a length of 1 unit.

$\mathbf{i}$ is the unit vector in the positive $x$ -axis direction (horizontal).
$\mathbf{j}$ is the unit vector in the positive $y$ -axis direction (vertical).

Vector $\overrightarrow{OQ}$ can be expressed as a combination of horizontal movement of $x$ and vertical movement of $y$ . In unit vector form, we write:

\overrightarrow{OQ} = x\mathbf{i} + y\mathbf{j}

Vector Components and Magnitude

The values $x$ and $y$ in vector $\overrightarrow{OQ} = x\mathbf{i} + y\mathbf{j}$ are called the vector components.

$x$ is the horizontal component. It's like the shadow of the vector on the $x$ -axis when illuminated from above.
$y$ is the vertical component. It's like the shadow of the vector on the $y$ -axis when illuminated from the side.

A vector with these two components is called a two-dimensional vector.

The length or magnitude of vector $\overrightarrow{OQ}$ , written as $|\overrightarrow{OQ}|$ , is the distance from the origin point $O$ to the endpoint $Q$ . If $Q(x,y)$ is the endpoint of the vector and $R(x,0)$ is the projection of point Q onto the $x$ -axis, we can calculate it using the Pythagorean theorem on the right-angled triangle $ORQ$ :

|\overrightarrow{OQ}| = \sqrt{x^2 + y^2}