1.2. Units Concepts http://www.ck12.org
Vector Addition and Subtraction
Like scalars, vectors have a branch of mathematics dedicated to them; and again, the basics can be considered an
extension of our natural abilties, while the more advanced parts are quite foreign to our intuition. The first concept
is that of vector addition. Think about throwing a pass, but this time to a moving target. If we use our original
arrow, the target will have moved by the time the ball reaches its endpoint. To be accurate, we need to consider the
displacement of the target and add it to the original arrow. The picture can be presented this way:
It should be apparent that if we throw the ball according to the dashed arrow, we will hit the target. This third vector
is the sum of the first two displacements, and of course, also a displacement vector. This is how vectors are added
graphically: if the end of the first vector is drawn at the beginning of the second, the arrow linking the beginning of
the first with the end of the second will be their sum. Alternatively, the two vectors can be moved to become the legs
of a parallelogram. Their sum is then the diagonal:
To subtract vectors, you can simply flip the vector you are subtracting by 180 degrees and add them. This is
essentially the vector version of saying that subtracting a positive number is the same as adding a negative one:
Vector Components
From the above examples, it should be clear that two vectors add to make another vector. Sometimes, the opposite
operation is useful: we often want to represent a vector as the sum of two other vectors. This is called breaking a
vector into its components. When vectors point along the same line, they essentially add as scalars. If we break
vectors into components along the same lines, we can add them by adding their components. The lines we pick to