Draw the two vectors head to tail. Then draw a vector from the tail of the first to the head of the second. This represents the resultant (Fig. 3.5).
head
tail FIG. 3.5 Combining vectors using the triangle law Combining three or more vectors
If we are combining three or more vectors, we must first find the resultant of any two vectors. Every time you do this, there will be one less vector. Eventually, you will have only two vectors to combine to a final resultant.
For example, there are three vectors to combine in Fig. 3.6a.
• First we use Pythagoras’s theorem to find the resultant of the two perpendicular 3 N vectors (Fig. 3.6b).
• This resultant is collinear with the 7 N vector (Fig. 3.6c), so we can simply add or subtract them.
• The final resultant is 2.76 N south-west, as shown in Fig. 3.6d. 3 N 135° 3 N 3 N 45° 135° 3 N 180° STS
Resolution of multiple vectors can be used to find the magnitude and direction of a force exerted by pulleys or wires in different systems. For example, radio/TV masts are held in place by stays or guys. The angle and tension of each stay must be calculated to keep the mast stable.
Even fixed structure towers need vector analysis to allow for weight distribution and design of cantilevers (load-carrying beams).
7 N a b FIG. 3.6 Combining three vectors Finding a resultant by experiment
Force is a vector quantity, with the newton as its unit. If three forces are keeping a particle at rest, the resultant of any two of those forces is equal in magnitude but opposite in direction to the third force. This fact can be used to find the resultant of two forces experimentally.
