Category: Vectors

There are three ways in which vectors can be multiplied, but none is exactly like the usual algebraic multiplication. As you read this section, keep in mind that a vector-capable calculator will help you multiply vectors only if you understand the basic rules of that multiplication. Multiplying a Vector by a Scalar If we multiply…

So far, in every figure that includes a coordinate system, the x and y axes are parallel to the edges of the book page. Thus, when a vector  is included, its components ax and ay are also parallel to the edges (as in Fig. 3-18a). The only reason for that orientation of the axes is that it looks “proper”; there is no deeper reason. We…

Using a sketch, we can add vectors geometrically. On a vector-capable calculator, we can add them directly on the screen. A third way to add vectors is to combine their components axis by axis, which is the way we examine here. To start, consider the statement which says that the vector  is the same as the…

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 , , and , where the hat ^ is used instead…

Adding vectors geometrically can be tedious. A neater and easier technique involves algebra but requires that the vectors be placed on a rectangular coordinate system. The x and y axes are usually drawn in the plane of the page, as shown in Fig. 3-8a. The z axis comes directly out of the page at the origin; we ignore it for now and…

Suppose that, as in the vector diagram of Fig. 3-2a, a particle moves from A to B and then later from B to C. We can represent its overall displacement (no matter what its actual path) with two successive displacement vectors, AB and BC. The net displacement of these two displacements is a single displacement from A to C. We call AC the vector sum (or resultant) of the vectors AB and BC. This sum is not the usual…

A particle moving along a straight line can move in only two directions. We can take its motion to be positive in one of these directions and negative in the other. For a particle moving in three dimensions, however, a plus sign or minus sign is no longer enough to indicate a direction. Instead, we…

Physics deals with a great many quantities that have both size and direction, and it needs a special mathematical language—the language of vectors—to describe those quantities. This language is also used in engineering, the other sciences, and even in common speech. If you have ever given directions such as “Go five blocks down this street…