0.9 Phy1060: motion -- displacement and vectors  (Page 5/13)

The displacement vector vecAC is the straight line from point A to point C, pointing in a direction from A to C. This vector is shown in the vector equation

vecAC = vecAB + vecBC

as the sum of the two vectors that describe the two segments of your journey from A to C.

Graphical addition of vectors

This process of connecting two or more vectors tail-to-head in sequence and then measuring the distance and direction from the tail of the first vector tothe head of the last vector is known as the graphical addition of vectors. The resulting vector (in this case vecAC) is known as the sum of the vectors thatmake up the sequence. (The resulting vector is often called the resultant.)

A 3-4-5 triangle

Recalling what we learned in an earlier module where I discussed a 3-4-5 triangle, the magnitude of the vector from A to C (the hypotenuse of a righttriangle with sides of 30 and 40) is 50 meters. I also recall that the angle is approximately 53 degrees relative to the horizontal (east-west) line.

You should be able to measure the length of vecAC as well as the angle between that vector and the horizontal and get reasonable agreement with theabove.

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.

Vectors on a graph board

As a blind student, you can't easily draw vectors, but you can represent vectors on your graph board by connecting two pushpins with something straight(like a stretched rubber band) and using something tactile to mark the head.

(Perhaps you could use a pushpin with a cylindrical top to indicate the tail and a pushpin with a spherical top to represent the head. Another option might be toget some small buttons and place the pin at the head through one of the holes in the button.)

Somewhat inconvenient

Constructing vectors on a graph board is less convenient for blind students than drawing them on graph paper is for sighted students. 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 choicefor blind students.

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.

One textbook describes vectors as "straight lines, with proper directions, indicated by arrowheads, and with lengths drawn to scale designated by vecAB (mynotation), etc.