Triangle Law, Parallelogram Law, and Scalar Multiplication

The following applet demonstrates geometrically the Triangle Law of addition and subtraction. It also demonstrates the commutative property of vectors with the Parallelogram Law and Scalar Multiplication. To view only the Triangle Law uncheck "Show Vector Subtraction" and "Show Parallelogram Law". The only point that you cannot move is point D. All other points may be moved as desired. To demonstrate scalar multiplication of a vector, uncheck all three check boxes. You will be vector D In the bottom input box, multiply vector DC by any number desired.

The yellow vectors represent the new vector obtained by adding the two red vectors or the two blue vectors or subtracting vector CE from vector DC. How are the vectors added Geometrically? How are they added algebraically?

Notice that vectors DC and BA have the same components even though they are in different locations. So do vectors CA and DB. This demonstrates that you can move vectors all around the coordinate plane and as long as they have the same direction and same magnitude, they are the same vector as they would have been if their starting point were at the origion.

What happens when you multiply vector DC by some number? Do you see why we name this Scalar Multiplication?

Jeremy Hidalgo, Created with GeoGebra