A vector can be represented by an arrow, drawn to some scale, whose length represents the magnitude of the vector and whose orientation in space represents the direction of the vector. The algebraic symbol for a vector quantity is set in boldface type.
In this figure, vector A is 6 units long at an angle alpha N of E (or in the "1st quadrant"), and vector B is 4 units long at an angle beta N of W (or in the "2nd quadrant").
The end of the arrow where the arrowhead is drawn is called the "tip" (or "head") of the vector, whereas the other end is called the "tail".
The vector sum of two or more vector quantities of the same kind is itself a vector, which is called the resultant R. For instance, the sum of vectors A and B is written symbolically as
Graphical Method ("Tail-to-Tip")
Consider the two vectors, A and B, discussed above. Their resultant can be found graphically by first redrawing the tail of B (maintaining its original length and direction) at the tip of A . The resultant R is then the vector drawn from the tail of A to the tip of B.
The magnitude of R is determined by measuring its length on the diagram, and compare it with the scale to which the vectors A and B were drawn. The direction of R with respect to the x-axis, which is represented by the angle theta, can be measured with a protractor.
The procedure just described can be extended to add more than two vectors. Place the tail of each vector at the tip of the previous vector, keeping their length and orientation unchanged, and draw the resultant vector R from the tail of the 1st vector to the tip of the last vector.
The graphical method for adding vectors may not be sufficiently accurate for problems to be solved, since it largely depends on how accurately the diagram is drawn. In the following, we will discussed an analytical method for finding the exact resultant (to the numerical accuracy of the calculating device used).
Analytical Method ("Rectangular Component Method")
This method employs exact equations that will enable us to find the magnitude and direction of the resultant of any number of vectors. To use this method, we first have to resolve each vector into its components.
The components of a vector are defined as the projections of the vector onto two arbitrary axes. It is usually convenient to determine the components relative to two perpendicular axes, usually chosen as the horizontal (x-axis) and the vertical (y-axis) directions.
For example, the vector A that makes an angle alpha with the x-axis has components Ax and Ay, respectively, along the x- and y- axes.
The original vector A and its components Ax and Ay form a right triangle whose hypotenuse is A and whose sides are Ax and Ay. Therefore, the magnitudes of Ax and Ay can be found analytically by using the trigonometry of a right triangle as: Ax = A cos(alpha), and Ay = A sin(alpha), where alpha is the angle that the vector A makes with the x-axis. Notice that the x- and y- components of a vector are signed numbers (positive or negative), the sign depending on which quadrant the angle lies in.
To determine the resultant vector R as the vector sum of two vectors, A and B, we first add the x-components of the vectors using ordinary arithmetic:
Since Rx is perpendicular to Ry, the magnitude of R is determined by using the Phytagorean theorem as
Vector quantities of any type can be added by either of the two methods discussed above. However, forces are the only vector quantity that can be added experimentally by use of a force table. Force can be thought of as any kind of push or pull. In this experiment, we will use the force of gravity, which in everyday's language is known as the "weight" of an object.
A force table consists of a level horizontal disk to which movable pulleys are attached.
The outer rim of the table is calibrated in degrees from 0 to 360. Strings are attached at one end to a ring at the center of the table. At the other end, the string passes over a pulley and hangs vertically. A slotted mass hanger is hung from each of the vertical strings. Forces are applied to the central ring by placing a set of mass on each mass holder. A removable pin within the prevents it from moving until all the forces are nearly balanced. The direction of the force vector is represented by clamping the pulley at the appropriate angle indicated on the circular scale.
The magnitude of each individual force applied to the ring is equal to the weight w of the set of mass placed on the mass holder, plus the mass of the holder. Since weight is defined as mass times acceleration of gravity g (=9.8 m/s2), the force in this experiment is given as
Several forces of known magnitude and direction can be simultaneously applied to the ring by placing the appropriate mass on a properly positioned mass holder. If the resultant force R is not zero, the ring will move when the pin is removed. In this event, we call the system is not in "equilibrium" condition.
To balance the system (means: to put the system in equilibrium condition), another force known as the equilibrant E, which is equal to (-)R must be added to the system, such that the entire sum of the forces is zero. [The vector E is a vector that is equal in magnitude to that of R, but opposite in direction from it; that is, E is 180 o from R.