Adding Vectors - Components of Vectors

Figure 3-7 (a) The components ax and ay of vector $\vec{a}$ . (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 $\vec{ba}$ 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, a_x and a_y are both positive because $\vec{a}$ extends in the positive direction of both axes. (Note the small arrowheads on the components, to indicate their direction.) If we were to reverse vector $\vec{a}$ , then both components would be negative and their arrowheads would point toward negative x and y. Resolving vector $\vec{b}$ in Fig. 3-8 yields a positive component b_x and a negative component b_y.

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 $\vec{a}$ in Fig. 3-7a geometrically from the right triangle there:

a_x = a cos θ and a_y + 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 $\vec{a}$ . Figure 3-7c shows that $\vec{a}$ 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 component 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, $\vec{a}$ in Fig. 3-7a is given (completely determined) by a and θ. It can also be given by its components a_x and a_y. Both pairs of values contain the same information. If we know a vector in component notation (a_x and a_y) and want it in magnitude-angle notation (a and θ), we can use the equations

$a = \sqrt{a_{x 2} + a_{y 2}}$ 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 (a_x, a_y, and a_z) to specify a vector.

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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