Adding and Subtracting Vectors Using Components

Using Components to Add and Subtract Vectors

Another way of adding vectors is to add the components. Previously, we saw that vectors can be expressed in terms of their horizontal and vertical components . To add vectors, merely express both of them in terms of their horizontal and vertical components and then add the components together.

Vector with Horizontal and Vertical Components

The vector in this image has a magnitude of 10.3 units and a direction of 29.1 degrees above the x-axis. It can be decomposed into a horizontal part and a vertical part as shown.

For example, a vector with a length of 5 at a 36.9 degree angle to the horizontal axis will have a horizontal component of 4 units and a vertical component of 3 units. If we were to add this to another vector of the same magnitude and direction, we would get a vector twice as long at the same angle. This can be seen by adding the horizontal components of the two vectors ($4+4$) and the two vertical components ($3+3$). These additions give a new vector with a horizontal component of 8 ($4+4$) and a vertical component of 6 ($3+3$). To find the resultant vector, simply place the tail of the vertical component at the head (arrow side) of the horizontal component and then draw a line from the origin to the head of the vertical component. This new line is the resultant vector. It should be twice as long as the original, since both of its components are twice as large as they were previously.

To subtract vectors by components, simply subtract the two horizontal components from each other and do the same for the vertical components. Then draw the resultant vector as you did in the previous part.