Vector Addition Using Components

Vector Addition Using Components

Components can also be used to find the resultant of vectors. This technique can be applied to both graphical and algebraic methods of finding the resultant. The method is straightforward:

1. make a rough sketch of the problem;
2. find the horizontal and vertical components of each vector;
3. find the sum of all horizontal components, \(\vec{R}_x\);
4. find the sum of all the vertical components, \(\vec{R}_y\);
5. then use them to find the resultant, \(\vec{R}\).

Consider the two vectors, \(\vec{F}_1\) and \(\vec{F}_2\), in the figure below, together with their resultant, \(\stackrel{\to }{R}\).

An example of two vectors being added to give a resultant.

Each vector in the figure above can be broken down into one component in the \(x\)-direction (horizontal) and one in the \(y\)-direction (vertical). These components are two vectors which when added give you the original vector as the resultant. This is shown in the figure below:

Adding vectors using components.

We can see that: \begin{align*} \vec{F}_1 &= \vec{F}_{1x} + \vec{F}_{1y} \\ \vec{F}_2 &= \vec{F}_{2x} + \vec{F}_{2y} \\ \vec{R} &= \vec{R}_x + \vec{R}_y \end{align*} \begin{align*} \text{But,}\ \vec{R}_x &= \vec{F}_{1x} + \vec{F}_{2x} \\ \text{and}\ \vec{R}_y &= \vec{F}_{1y} + \vec{F}_{2y} \end{align*}

In summary, addition of the \(x\)-components of the two original vectors gives the \(x\)-component of the resultant. The same applies to the \(y\)-components. So if we just added all the components together we would get the same answer! This is another important property of vectors.

Optional Video on Vectors