2.6 Game 2302-0360 motion -- displacement and vectors  (Page 4/11)

No requirement for a right triangle

While this example is based on a right triangle, that is not a requirement for adding vectors. I chose a 3-4-5 right triangle for the examplebecause I knew what the answer would be in advance and also because it should have been familiar to you.

Graphic representation of vectors

Many problems in kinematics and kinetics can be solved graphically using a graphical representation of vectors.

Vectors are typically drawn as a heavy line between two points with an arrow head at one end and nothing in particular at the other end. The end with thearrow head is commonly called the head of the vector and the other end is commonly called the tail.

Somewhat inconvenient

Drawing vectors on graph paper can be inconvenient, particularly if you don't have graph paper available. Fortunately, as you willsoon learn, it isn't necessary to use graphics to solve vector problems. You can also solve such problems mathematically, which will often be the better choice.

Addition and subtraction of vectors

There are at least two kinds of quantities in physics:

• scalars , having magnitude only
• vectors , having both magnitude and direction

You already know how to do scalar arithmetic. Otherwise, you probably wouldn't be interested in physics.

Another example

Let's go back to our original equation

vecAC = vecAB + vecBC

and assume that the magnitude of vecAB is 30 and the magnitude of vecBC is 40. The sum of those two vectors (vecAC) can have a magnitude ranging anywhere from 10 to 70depending on the relative angles of vecAB and vecBC.

A triangle with sides of 30, 4 0, and ?

Consider the triangle shown in Figure 3 .

Figure 3 - A triangle with sides of 30, 40, and ?.

Pretend that instead of walking due north from point B as shown in Figure 2 , you change direction and walk northwest (135 degrees relative to the east-west horizontal line with east being zerodegrees) keeping the length at 40 meters.

What happened to the displacement?

What is the displacement of the point C relative to the point A? I can't do the arithmetic in my head for this problem, but I can measure the length of vecAC to be about 28 meters and the angle of vecAC to be a little less than 90degrees. (I will show you how to write a script to solve this problem mathematically later.)

Any number of displacements can be added

There is no limit to the number of displacement vectors that can be added in this manner. For example, pretend that you walk from point A,

• 10 meters east to point B,
• 12 meters southwest to point C,
• 30 meters north to point D,
• 15 meters east to point E,
• 35 meters southwest to point F, and
• 44 meters north to point G where you stop.

Your displacement vecAG will be

vecAG = vecAB + vecBC + vecCD + vecDE + vecEF + vecFG

even if your zigzag path crosses back over itself one or more times.

Vector diagram for the sum of six vectors

Figure 4 shows the graphical addition of the six vectors described above.

Figure 4 - Vector diagram for the sum of six vectors.

The displacement in Figure 4 is the vector with its tail at A and its head at G, which you could measure with a measuring stick anda protractor.