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

Figure 3.13 Unit vectors $\hat{i}$, $\hat{j}$, and $\hat{k}$, 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 $\hat{i}$, $\hat{j}$, and $\hat{k}$, 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 $\overrightarrow{a}$ and $\overrightarrow{a}$ of Figs. 3-7 and 3-8 as

$\overrightarrow{a}=a_x\hat{i}+a_y\hat{j}$(3-7)

and $\overrightarrow{b}=b_x\hat{i}+b_y\hat{j}$(3-8)

These two equations are illustrated in Fig. 3-14. The quantities $a_x\hat{i}$ and $a_y\hat{j}$ are vectors, called the vector components of $\overrightarrow{a}$. The quantities a_x and a_y are scalars, called the scalar components of $\overrightarrow{a}$ (or, as before, simply its a components).

Adding Vectors by 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

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

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

r_x = a_x + b_x (3-10)
r_y = a_y + b_y (3-11)
r_z = a_z + b_z (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 $\overrightarrow{a}$ and $\overrightarrow{b}$, 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 $\overrightarrow{r}$; and (3) combine the components of to get itself. We have a choice in step 3. We can express $\overrightarrow{r}$ 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 $\overrightarrow{d}=\overrightarrow{a}-\overrightarrow{b}$ can be rewritten as an addition $\overrightarrow{d}=\overrightarrow{a}+\overrightarrow{\mathrm{-b}}$. To subtract, we add and by components, $\overrightarrow{a}+\overrightarrow{\mathrm{-b}}$ to get d_x = a_x - b_x, d_y = a_y - b_y, and d_z = a_z - b_z, where $\overrightarrow{d}=d{x}_{}\hat{i}+d{y}_{}\hat{j}+d{z}_{}\hat{k}$.(3-13)

About the Authors

David Halliday was an American physicist known for his physics textbooks, Physics and Fundamentals of Physics, which he wrote with Robert Resnick. Both textbooks have been in continuous use since 1960 and are available in more than 47 languages.

Robert Resnick was a physics educator and author of physics textbooks. He was born in Baltimore, Maryland on January 11, 1923 and graduated from the Baltimore City College high school in 1939. He received his B.A. in 1943 and his Ph.D. in 1949, both in physics from Johns Hopkins University.

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