Graphical Addition

Consider the vectors u = (3, 4) and v = (4, 1) in the plane. From the component method of vector addition we know that the sum of these two vectors is u + v = (7, 5). Graphically, we see that this is the same as the result we would get by "picking up" one of the vectors (without changing either its direction or its magnitude), placing its end at the other (unmoved) vector's tip, and drawing an arrow from the origin to the new tip location for the displaced vector.

Figure %: The sum of the vectors u = (3, 4) and v = (4, 1) in the plane.

This geometric procedure for adding vectors works in general. For any two vectors u and v in the plane, the sum of the vectors is graphically given as in the following figure:

Figure %: The sum of the vectors u and v in the plane.
The geometric procedure is valid for 3-dimensional vectors as well. Notice that in the same way that any two lines lie in a plane, any two vectors in 3-dimensional space will also lie in the same plane. This recognition allows us to see that the sum of two vectors will always lie in the plane defined by the original two vectors.

As we noted in Vector Subtraction, in order to subtract one vector from another, you simply add its negative partner: u - v=u + (- 1)v. Thus, vectors can be subtracted graphically in the same manner used for adding them, by simply taking care to reverse the direction of the vector being subtracted:

Figure %: The difference of the vectors u and v in the plane.
If you graphically add back in the subtracted vector to your result from the subtraction and you recover the initial vector you subtracted from. In other words, (u - v) + v = u in our graphical methods, as we should expect!

Scalar Multiplication

What happens graphically when we multiply a vector by a scalar? The vector changes in length, while its direction remains the same. If the vector's magnitude was previously | v|, once it is multiplied by a scalar we have | av| = a| v|. Note that if | a| > 1 the new vector will be longer. If | a| < 1 the new vector will be shorter. And if a < 0, the new vector will point in the opposite direction as the original one.