# Adding Vectors - Components of Vectors by David Halliday , Robert Resnick

Figure 3-7 (a) The components ax and ay of vector $\stackrel{\to }{a}$. (b) The components are unchanged if the vector is shifted, as long as the magnitude and orientation are maintained. (c) The components form the legs of a right triangle whose hypotenuse is the magnitude of the vector.

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, in Fig. 3-7a.The z axis comes directly out of the page at the origin; we ignore it for now and deal only with two-dimensional vectors.

A component of a vector is the projection of the vector on an axis. In Fig. 3-7a, for example, ax is the component of vector on (or along) the x axis and ay is the component along the y axis. To find the projection of a vector along an axis, we draw perpendicular lines from the two ends of the vector to the axis, as shown.The projection of a vector on an x axis is its x component, and similarly the projection on the y axis is the y component. The process of finding the components of a vector is called resolving the vector.

Figure 3-8 The component of $\stackrel{\to }{\mathrm{ba}}$ on the x axis is positive, and that on the y axis is negative.

A component of a vector has the same direction (along an axis) as the vector. In Fig. 3-7, ax and ay are both positive because $\stackrel{\to }{a}$ extends in the positive direction of both axes. (Note the small arrowheads on the components, to indicate their direction.) If we were to reverse vector $\stackrel{\to }{a}$, then both components would be negative and their arrowheads would point toward negative x and y. Resolving vector $\stackrel{\to }{b}$ in Fig. 3-8 yields a positive component bx and a negative component by.

In general, a vector has three components, although for the case of Fig. 3-7a the component along the z axis is zero. As Figs. 3-7a and b show, if you shift a vector without changing its direction, its components do not change.

Finding the Components. We can find the components of $\stackrel{\to }{a}$ in Fig. 3-7a geometrically from the right triangle there:

ax = a cos θ and ay + a sin θ, (3-5)

where θ is the angle that the vector makes with the positive direction of the x axis, and a is the magnitude of $\stackrel{\to }{a}$. Figure 3-7c shows that $\stackrel{\to }{a}$ and its x and y components form a right triangle. It also shows how we can reconstruct a vector from its components: we arrange those components head to tail. Then we complete a right triangle with the vector forming the hypotenuse, from the tail of one component to the head of the other component.

Once a vector has been resolved into its components along a set of axes, the components themselves can be used in place of the vector. For example, $\stackrel{\to }{a}$ in Fig. 3-7a is given (completely determined) by a and θ. It can also be given by its components ax and ay. Both pairs of values contain the same information. If we know a vector in component notation (ax and ay) and want it in magnitude-angle notation (a and θ), we can use the equations

$a=\sqrt{{{a}_{x}}^{}+{{a}_{y}}^{}}$ and tan θ $=\frac{{a}_{y}}{{a}_{x}}$

to transform it.

In the more general three-dimensional case, we need a magnitude and two angles (say, a, θ, and Θ) or three components (ax, ay, and az) to specify a vector.