Vector Components

Why Vectors Are Split into Components

A slanted vector is difficult to calculate with if we only see it as one arrow. To make calculations easier, the vector is split into parts parallel to the coordinate axes. These parts are called vector components.

The horizontal component shows how much of the vector lies along the $x$ axis. The vertical component shows how much lies along the $y$ axis. In three-dimensional space, there is also a component along the $z$ axis.

Vector components turn one slanted arrow into parts that follow the axes.

Reading Components in a Plane

In a plane, vector $\vec{A}$ can be written as the combination of its $x$ and $y$ components.

\vec{A}=A_x\hat{i}+A_y\hat{j}

The notation $\hat{i}$ is the unit vector in the $x$ direction, while $\hat{j}$ is the unit vector in the $y$ direction. The numbers $A_x$ and $A_y$ give the component sizes along each direction.

For example, $\vec{A}=6\hat{i}+8\hat{j}$ means the vector moves $6$ units in the positive $x$ direction and $8$ units in the positive $y$ direction.

\lvert\vec{A}\rvert=\sqrt{6^2+8^2}=10

The vector length comes from the Pythagorean theorem because the $x$ and $y$ components are perpendicular.

Components in Three-Dimensional Space

In three-dimensional space, a vector needs three components. The general form is:

\vec{A}=A_x\hat{i}+A_y\hat{j}+A_z\hat{k}

The unit vector $\hat{k}$ points along the $z$ axis. This component is needed when motion or force is not limited to a flat plane, such as a drone moving east, north, and upward at the same time.

Components of

\vec{A}

Vector

\vec{A}=4\hat{i}+3\hat{j}+2\hat{k}

is split into three perpendicular components.

The visual uses component steps in order: move along the $x$ axis, then the $y$ axis, then the $z$ axis. The final endpoint is the same as the endpoint of vector $\vec{A}$ .

Positive and Negative Component Signs

Components can be positive or negative. The sign shows direction relative to the reference axis, not a negative vector magnitude.

Component	Positive sign means	Negative sign means
$A_x$	toward positive $x$	toward negative $x$
$A_y$	toward positive $y$	toward negative $y$
$A_z$	toward positive $z$	toward negative $z$

If $\vec{B}=-5\hat{i}+2\hat{j}$ , the vector has a $5$ unit component toward negative $x$ and a $2$ unit component toward positive $y$ .

Displacement Written as Components

A student walks $3\text{ m}$ east, then $4\text{ m}$ north. If east is the positive $x$ direction and north is the positive $y$ direction, the displacement is:

\vec{s}=3\hat{i}+4\hat{j}

Components keep the calculation organized because each direction is handled on its own axis.