Every vector can be thought of as the sum of two or more other vectors.
Consider the three vectors in the figure above. Since is composed of and the following expression is established:
Vector decomposition is the general process of breaking one vector into two or more vectors that add up to the original vector. The component vectors into which the original vector is decomposed are chosen based on specific details of the problem at hand.
Given the points and what is the value of where and satisfy the following equation:
Observe that was decomposed into three vectors that are parallel to the three coordinate axes. Also notice that the vectors and are the unit vectors of the coordinate space.
If and how can be expressed using and
Let Then we have
If the vectors and are parallel, what is the value of
Since and are parallel, a certain real number which satisfies exists. Thus,
The coordinates of points and are and respectively. If what are the coordinates of point
Let the position vector of point be Then we have
Since we are given we have
Thus, the coordinates of point are
If the vectors and satisfy and what is
One generally useful decomposition is with components that lie parallel to each of the coordinate axes.
The figure above shows the decomposition of a vector in a three-dimensional space. Observe that the red vector can be decomposed into the three green vectors that are parallel to the three coordinate axes. Thus, is composed of the three vectors and Therefore the following expression corresponds:
One can define the unit vectors that point in each of the positive coordinate axes as follows:
It follows that for any vector , it holds that
In this way, one frequently encounters vectors written in terms of the unit vectors , , and . An alternate notation uses , , and .
For instance, the vector can be expressed as .
Note that the coordinate unit vectors are orthonormal. That is to say, and .
When two vectors and add,
which is the same thing as . Similarly, when one takes the dot product , one obtains