## Vector Properties

Two vectors

aandbareequalif they have the samedirectionand the samemagnitude(or length). We could then writea = bA vector

amay be multiplied by a scalar s (remember, a scalar is an ordinary number). We could write this new vector asc= saThis new vector

chas the same direction as vectoraand its magnitude is s times the magnitude ofa.We write the

magnitude(or length) of a vector withoutboldface or withouta vector over it. The magnitude of a vector is an ordinary scalar; there is no direction associated with the magnitude of a vector.Speed is the magnitude of velocity and distance is the magnitude of displacement.

Consider two vectors

AandBwhich we want to add. They might be displacement vectors or velocity vectors or electric field vectors -- or any vectors at all. We can add themgraphicallyby drawing vectorAand then, at the tip of vectorA, drawing vectorBas shown below. The sum of vectors is called theresultant. TheresultantvectorR, is the vector that we can draw from the beginning ofAto the end ofB. We can write this asR = A + BVector addition is

commutative. That means theorderin which we add vectors does not affect the resultant. To add vectorsAandBwe could begin by drawing vectorB. At the end of vectorBwe would then draw vectorA. The resultant vectorRis then the vector we can draw by starting at the beginning ofBand finishing at the end ofA. We could write this asR = B + AThese resultants are exactly the same. That means

A + B = B + ASometime people will draw the vectors in twice to form a parallelogram as shown below. The resultant

Ris the diagonal as shown. This is referred to as vector addition by the parallelogram method.

My favorite description of

vector additioninvolves the pieces of an ancient treasure map. Suppose we find these fragments or pieces of an old treasure map. "the old oak tree" becomes the origin of our coordinate system.Each of these pieces of the map -- each of these statments of distance and direction -- may be represented by a vector:

If we follow these directions

in this order,we walk along the route given byR = A + B + C + DThe resultant vector

Rlocates where we end up after following these directions in this order. As we will see, this resultant isR = 11.2 paces at 27^{o}west of north.But the pieces of the treasure map may be shuffled. That is, we may follow the directions in a different order. Here is a vector addition diagram for

R = B + C + D + AWe still end up at the same spot. The resultant is independent of the order in which we add the vectors.

We might shuffle the pieces of the treasure map yet again and follow their directions in still a different order. Here is a vector addition diagram for

R = B + A + D + CFor these three examples, and all the other possible permutations, the resultant

Rwill always be the same. To determineRwe can carefully construct one of these drawingsto scaleand then carefully measure the length ofRand the direction ofR. When we do this we will findR = 11.2 paces at 27^{o}west of north

Vectors and Scalars Components Return to ToC, Vectors and 2D Motion(c) 2005, Doug Davis; all rights reserved