2.6 Game 2302-0360 motion -- displacement and vectors  (Page 5/11)

Can be extended to three (or more) dimensions

The physics textbook titled College Physics by Mendenhall, Eve, Keys, and Sutton contains the following example.

"If a man climbs up the mast of a ship sailing east while the tide caries it south, then that sailor may have three displacements at right angles, vecAB30 feet upward, vecBC 100 feet eastward, vecCD 20 feet southward, all with respect to the bed of the ocean. The three displacements vecAB, vecBC, and vecCDare equivalent to the single displacement vecAD; and in the same way, any number of displacements may be added together."

Difficult to draw

Of course, we can't easily draw that 3D vector diagram on a flat sheet of paper or construct it on a flat computer monitor, but we can solve for the displacement vecAD if we are familiar with computations in three dimensions. (An explanationof 3D computations is beyond the scope of this module.)

The parallelogram law

There is a law that states:

The sum of two vectors in a plane is represented by the diagonal of a parallelogram whose adjacent sides represent the two vector quantities.

The resultant

As I mentioned earlier, the sum of two or more vectors is called their resultant .

In general, the resultant is not simply the algebraic or arithmetic sum of the vector magnitudes. Instead, in our original vector equation

vecAC = vecAB + vecBC

vecAC is the resultant of the sum of vecAB and vecBC in the sense that a single vector vecAC would have the same effect as vecAB and vecAD actingjointly.

Geometrical addition

The parallelogram law is a form of geometrical addition. In engineering and physics, it isoften used to find graphical solutions to problems involving forces, velocities, displacements, accelerations, electric fields, and other directed quantities. (Directed quantities have both magnitude and direction.)

Solving a vector problem with a parallelogram

Pretend that you walk from point A

• 5 meters east to point B, followed by
• 6 meters at an angle of 30 degrees (north of east) to point C.

Let's use a vector diagram and the parallelogram law to find the resultant displacement vector vecAC.

Figure 5 illustrates the use of the parallelogram law to find a graphical solution to this problem.

Figure 5 - Illustration of the parallelogram law.

Put both vector tails at point A

Mentally designate the starting point on your vector diagram as point A. Construct a vector with a length of 5 meters and an angle of 0 degrees relativeto the horizontal with its tail at point A. Mentally designate this vector as vecAB.

Construct a second vector with a length of 6 meters at an angle of 30 degrees with its tail at point A. Mentally designate this vector as vecBC.

Parallel lines

Construct a line at least 6 meters long starting at the head of vecBC. Make this lineparallel to vecAB.

(There are drawing tools that make it easy to draw a linethat is parallel to another line. This is one reason why this is a popular technique for graphically adding vectors.)

Construct another line, at least 7 meters long, starting at the head of vecAB. Make thisline parallel to vecBC.