Vectors in Two Dimensions

Describing a Vector with Horizontal and Vertical Steps

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 directed line segment from the origin $O$ to another point, for example $Q(x, y)$ , we obtain a vector. This vector is written as $\overrightarrow{OQ}$ . Its magnitude is the segment's length, and its direction runs from the initial point toward the arrowhead.

Two unit vectors describe the coordinate directions. Each has a length of $1 \text{ unit}$ .

$\mathbf{i}$ is the unit vector along the positive $x$ -axis (horizontal).
$\mathbf{j}$ is the unit vector along the positive $y$ -axis (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 scalar component. The corresponding vector projection onto the $x$ -axis is $x\mathbf{i}$ .
$y$ is the vertical scalar component. The corresponding vector projection onto the $y$ -axis is $y\mathbf{j}$ .

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 $O$ to the endpoint $Q$ . If $Q(x,y)$ is the endpoint and $R(x,0)$ is the projection of $Q$ onto the $x$ -axis, the right triangle $ORQ$ gives its magnitude through the Pythagorean theorem:

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

Two Dimensional Vector Visualization

The visualization below connects the vector, its components, and its magnitude in the Cartesian plane:

Vector in the Coordinate Plane

Visualization of vector

OQ = 4\mathbf{i} + 3\mathbf{j}

in the Cartesian coordinate system

The diagram shows all three coordinate axes. The vector lies in the $xy$ -plane, so its $z$ -component is $0$ .

In the visualization above:

Vector $\overrightarrow{OQ}$ (light green) has initial point $O(0,0)$ and endpoint $Q(4,3)$
The $x$ -component (light blue) is the projection of vector $\overrightarrow{OQ}$ on the $x$ -axis, which is $4\mathbf{i}$
The $y$ -component (yellow) is the projection of vector $\overrightarrow{OQ}$ on the $y$ -axis, which is $3\mathbf{j}$
The magnitude of the vector $|\overrightarrow{OQ}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5$

Describing a Vector with Horizontal and Vertical Steps

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 directed line segment from the origin

O

to another point, for example

Q(x, y)

, we obtain a vector. This vector is written as

\overrightarrow{OQ}

. Its magnitude is the segment's length, and its direction runs from the initial point toward the arrowhead.

Two unit vectors describe the coordinate directions. Each has a length of

1 \text{ unit}

\mathbf{i}

is the unit vector along the positive

x

-axis (horizontal).

\mathbf{j}

is the unit vector along the positive

y

-axis (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 scalar component. The corresponding vector projection onto the

x

-axis is

x\mathbf{i}

y

is the vertical scalar component. The corresponding vector projection onto the

y

-axis is

y\mathbf{j}

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

O

to the endpoint

Q

. If

Q(x,y)

is the endpoint and

R(x,0)

is the projection of

Q

onto the

x

-axis, the right triangle

ORQ

gives its magnitude through the Pythagorean theorem:

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

Two Dimensional Vector Visualization

The visualization below connects the vector, its components, and its magnitude in the Cartesian plane:

Vector in the Coordinate Plane

Visualization of vector

OQ = 4\mathbf{i} + 3\mathbf{j}

in the Cartesian coordinate system

The diagram shows all three coordinate axes. The vector lies in the

xy

-plane, so its

z

-component is

0

In the visualization above:

Vector

\overrightarrow{OQ}

(light green) has initial point

O(0,0)

and endpoint

Q(4,3)

The

x

-component (light blue) is the projection of vector

\overrightarrow{OQ}

on the

x

-axis, which is

4\mathbf{i}

The

y

-component (yellow) is the projection of vector

\overrightarrow{OQ}

on the

y

-axis, which is

3\mathbf{j}

The magnitude of the vector

|\overrightarrow{OQ}| = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5