2.1 Vectors in the plane  (Page 2/29)

Combining vectors

Vectors have many real-life applications, including situations involving force or velocity. For example, consider the forces acting on a boat crossing a river. The boat’s motor generates a force in one direction, and the current of the river generates a force in another direction. Both forces are vectors. We must take both the magnitude and direction of each force into account if we want to know where the boat will go.

A second example that involves vectors is a quarterback throwing a football. The quarterback does not throw the ball parallel to the ground; instead, he aims up into the air. The velocity of his throw can be represented by a vector. If we know how hard he throws the ball (magnitude—in this case, speed), and the angle (direction), we can tell how far the ball will travel down the field.

A real number is often called a scalar    in mathematics and physics. Unlike vectors, scalars are generally considered to have a magnitude only, but no direction. Multiplying a vector by a scalar changes the vector’s magnitude. This is called scalar multiplication. Note that changing the magnitude of a vector does not indicate a change in its direction. For example, wind blowing from north to south might increase or decrease in speed while maintaining its direction from north to south.

As you might expect, if $k=\mathrm{-1},$ we denote the product $k\text{v}$ as

Note that $\text{−}\text{v}$ has the same magnitude as $\text{v},$ but has the opposite direction ( [link] ).

Another operation we can perform on vectors is to add them together in vector addition, but because each vector may have its own direction, the process is different from adding two numbers. The most common graphical method for adding two vectors is to place the initial point of the second vector at the terminal point of the first, as in [link] (a). To see why this makes sense, suppose, for example, that both vectors represent displacement. If an object moves first from the initial point to the terminal point of vector $\text{v},$ then from the initial point to the terminal point of vector $\text{w},$ the overall displacement is the same as if the object had made just one movement from the initial point to the terminal point of the vector $v+w.$ For obvious reasons, this approach is called the triangle method    . Notice that if we had switched the order, so that $\text{w}$ was our first vector and v was our second vector, we would have ended up in the same place. (Again, see [link] (a).) Thus, $v+w=w+v.$

A second method for adding vectors is called the parallelogram method    . With this method, we place the two vectors so they have the same initial point, and then we draw a parallelogram with the vectors as two adjacent sides, as in [link] (b). The length of the diagonal of the parallelogram is the sum. Comparing [link] (b) and [link] (a), we can see that we get the same answer using either method. The vector $v+w$ is called the vector sum    .