Vector addition

Suppose that the vector displacement ${\bf r}$ of some point $R$ from the origin $O$ is specified as follows:

$${\bf r} = {\bf r}_1 + {\bf r}_2.$$ (31)

Figure 12 illustrates how this expression is interpreted diagrammatically: in order to get from point $O$ to point $R$, we first move from point $O$ to point $S$ along vector ${\bf r}_1$, and we then move from point $S$ to point $R$ along vector ${\bf r}_2$. The net result is the same as if we had moved from point $O$ directly to point $R$ along vector ${\bf r}$. Vector ${\bf r}$ is termed the resultant of adding vectors ${\bf r}_1$ and ${\bf r}_2$.

Figure 12: Vector addition

Note that we have two ways of specifying the vector displacement of point $S$ from the origin: we can either write ${\bf r}_1$ or ${\bf r} - {\bf r}_2$. The expression ${\bf r} - {\bf r}_2$ is interpreted as follows: starting at the origin, move along vector ${\bf r}$ in the direction of the arrow, then move along vector ${\bf r}_2$ in the opposite direction to the arrow. In other words, a minus sign in front of a vector indicates that we should move along that vector in the opposite direction to its arrow.

Suppose that the components of vectors ${\bf r}_1$ and ${\bf r}_2$ are $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$, respectively. As is easily demonstrated, the components $(x,y,z)$ of the resultant vector ${\bf r}= {\bf r}_1 + {\bf r}_2$ are

$$x = x_1+x_2,$$ (32)

$$y = y_1 + y_2,$$ (33)

$$z = z_1 + z_2.$$ (34)

In other words, the components of the sum of two vectors are simply the algebraic sums of the components of the individual vectors.