How to Find a Resultant Vector by Components

A resultant vector represents the combined effect of several vectors. Component addition is a reliable method because it works for two vectors or many vectors and does not depend on drawing them perfectly to scale.

Choose axes and signs first

Select a coordinate system before calculating. In two dimensions, a common choice is positive $x$ to the right and positive $y$ upward. Resolve every vector into components in that same system:

$$\mathbf A=A_x\hat{\mathbf i}+A_y\hat{\mathbf j}.$$

Signs encode direction. A leftward component is negative when right is positive; a downward component is negative when up is positive.

Add like components

For any number of vectors,

$$R_x=\sum_i A_{i,x},\qquad R_y=\sum_i A_{i,y}.$$

Then

$$\mathbf R=R_x\hat{\mathbf i}+R_y\hat{\mathbf j}.$$

Do not add magnitudes unless all vectors lie along the same line and their directions have already been represented by signs.

A new example

Suppose one force is $6\,\text{N}$ east and another is $8\,\text{N}$ north. Their component sum is

$$\mathbf R=(6\hat{\mathbf i}+8\hat{\mathbf j})\,\text{N}.$$

The magnitude is

$$|\mathbf R|=\sqrt{6^2+8^2}=10\,\text{N}.$$

Its direction measured north of east is

$$\theta=\tan^{-1}\!\left(\frac{8}{6}\right)\approx53.1^\circ.$$

So the resultant is $10\,\text{N}$ at $53.1^\circ$ north of east.

Check the result

The units of every component and the resultant must match. The signs should place the direction in the correct quadrant. Also check the triangle inequality: the resultant magnitude cannot exceed the sum of the input magnitudes. If all component sums are zero, the resultant is the zero vector and has no unique direction.