Vector Addition
There are two main operations on vectors. The first one is the addition of two vectors. Let's do the following in the applet below:
- Construct two vectors u and v in using the vector tool
. - Drag u to the origin i.e. the tail of u is at the origin.
- Drag v to the arrowhead of u.
- The vector u + v is defined as the vector pointing from the origin to the arrowhead of v.
Alternatively, you can regard the vector u + v as the "diagonal vector" of the parallelogram formed by the two vectors u and v pointing out from the origin. For the physics viewpoint, this definition of addition is quite natural. You can imagine two forces represented by u and v act on a mass at the origin. The resultant force is exactly u + v.
Draw three vectors u, v, and w in the applet. What can you say about the vectors u + v, v + u, (u + v) + w, and u + (v + w) ?
How are the column vectors u and v related to the column vector u + v ? Explain your answer briefly.
Higher-Dimensional Vectors
The definition of addition for higher-dimensional vectors is analogous to the one for vectors in or . It is harder to visualize the addition procedure in higher-dimensional space. Nonetheless, we can define the vector addition by adding the corresponding column vectors in a similar way.
The following are the random examples of the vector addition of two n-dimensional vectors (expressed as column vectors).