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Section 1.4 Vectors

We begin with the intuitive idea that a vector \(\ww\) is an arrow in space. Examples of vectors include the displacement from one point to another and your velocity at a point as you are moving along some path. An explicit example is shown in Figure 1.6.

Figure 1.6. A vector \(\ww\text{.}\)

What operations can we do with vectors? Two vectors can be added, using the parallelogram rule, as shown in Figure 1.7.

Figure 1.7. Adding two vectors using the parallelogram rule.

Vectors in space are anchored with their tails at a point in space, as shown by the dot in the figures. As shown in Figure 1.7, one adds vectors that are anchored at the same point, and the result is another vector anchored at that point.

Once we know how to add vectors, we can also rescale them. For instance, \(2\ww=\ww+\ww\text{.}\) This operation can be generalized to rescale \(\ww\) by any real number, a process known as scalar multiplication.

Defining \(-\ww\) to point in the opposite direction from \(\ww\text{,}\) we have

\begin{equation} \ww-\ww = \ww+(-\ww) = \zero\tag{1.4.1} \end{equation}

which could also have been taken as the definition of \(-\ww\text{.}\) Thus, we can also subtract vectors, as shown in Figure 1.8.

Figure 1.8. Subtracting two vectors using the parallelogram rule.