# Vectors - Unit Vectors and Adding Vectors by Componets by David Halliday , Robert Resnick

Figure 3.13 Unit vectors $\stackrel{^}{i}$, $\stackrel{^}{j}$, and $\stackrel{^}{k}$, define the directions of a right-handed coordinate system.

A unit vector is a vector that has a magnitude of exactly 1 and points in a particular direction. It lacks both dimension and unit. Its sole purpose is to point - that is, to specify a direction. The unit vectors in the positive directions of the x, y, and z axes are labeled $\stackrel{^}{i}$, $\stackrel{^}{j}$, and $\stackrel{^}{k}$, where the hat is used instead of an overhead arrow as for other vectors (Fig. 3-13). The arrangement of axes in Fig. 3-13 is said to be a right-handed coordinate system. The system remains right-handed if it is rotated rigidly. We use such coordinate systems exclusively in this book.

Unit vectors are very useful for expressing other vectors; for example, we can express $\stackrel{\to }{a}$ and $\stackrel{\to }{a}$ of Figs. 3-7 and 3-8 as

$\stackrel{\to }{a}={a}_{x}\stackrel{^}{i}+{a}_{y}\stackrel{^}{j}$(3-7)

and $\stackrel{\to }{b}={b}_{x}\stackrel{^}{i}+{b}_{y}\stackrel{^}{j}$(3-8)

These two equations are illustrated in Fig. 3-14. The quantities ${a}_{x}\stackrel{^}{i}$ and ${a}_{y}\stackrel{^}{j}$ are vectors, called the vector components of $\stackrel{\to }{a}$. The quantities ax and ay are scalars, called the scalar components of $\stackrel{\to }{a}$ (or, as before, simply its a components).

We can add vectors geometrically on a sketch or directly on a vector-capable calculator.A third way is to combine their components axis by axis.

To start, consider the statement

$\stackrel{\to }{r}=\stackrel{\to }{a}+\stackrel{\to }{b}$(3-9)

which says that the vector $\stackrel{\to }{r}$ is the same as the vector $\stackrel{\to }{a}+\stackrel{\to }{b}$. Thus, each component of $\stackrel{\to }{r}$ must be the same as the corresponding component of $\stackrel{\to }{a}+\stackrel{\to }{b}$:

rx = ax + bx (3-10)
ry = ay + by (3-11)
rz = az + bz (3-12)

In other words, two vectors must be equal if their corresponding components are equal. Equations 3-9 to 3-12 tell us that to add vectors $\stackrel{\to }{a}$ and $\stackrel{\to }{b}$, we must (1) resolve the vectors into their scalar components; (2) combine these scalar components, axis by axis, to get the components of the sum $\stackrel{\to }{r}$; and (3) combine the components of to get itself. We have a choice in step 3. We can express $\stackrel{\to }{r}$ in unit-vector notation or in magnitude-angle notation.

This procedure for adding vectors by components also applies to vector subtractions. Recall that a subtraction such as $\stackrel{\to }{d}=\stackrel{\to }{a}-\stackrel{\to }{b}$ can be rewritten as an addition $\stackrel{\to }{d}=\stackrel{\to }{a}+\stackrel{\to }{\mathrm{-b}}$. To subtract, we add and by components, $\stackrel{\to }{a}+\stackrel{\to }{\mathrm{-b}}$ to get dx = ax - bx, dy = ay - by, and dz = az - bz, where $\stackrel{\to }{d}=d{x}_{}\stackrel{^}{i}+d{y}_{}\stackrel{^}{j}+d{z}_{}\stackrel{^}{k}$.(3-13)