Vector Components

Why Vectors Are Split into Components

A slanted vector acts along more than one coordinate axis. We split it into parts parallel to those axes so that the effect in each direction can be calculated separately. 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.

A vector equals the sum of its components along the coordinate 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 signed scalars $A_x$ and $A_y$ give the components along those axes. Each sign records which way along its axis the vector points.

For example, $\vec{A}=6\hat{i}+8\hat{j}$ means the vector has a $6$ unit component in the positive $x$ direction and an $8$ unit component 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

The vector's magnitude stays zero or positive. A component's sign marks the direction along its axis.

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}

With components, the horizontal and vertical directions are calculated on their own axes.

The component form shows a net displacement toward the east and north. Its magnitude $5\text{ m}$ comes from a right triangle with side lengths $3\text{ m}$ and $4\text{ m}$ .