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The vector Figure 3.1.13
is the difference of vectors and , so
EXAMPLE 2 Finding the Components of a Vector
The components of the vector with initial point and terminal point are
In 2-space the vector with initial point and terminal point is
Translation of Axes
The solutions to many problems can be simplified by translating the coordinate axes to obtain new axes parallel to the original
ones.
In Figure 3.1.14a we have translated the axes of an -coordinate system to obtain an -coordinate system whose origin is
coordinates. To see how the two are
at the point . A point P in 2-space now has both (x, y) coordinates and
related, consider the vector (Figure 3.1.14b). In the -system its initial point is at k, l) and its terminal point is at (x), (y), so
. In the -system its initial point is at (0, 0) and its terminal point is at, , so .
Therefore,
These formulas are called the translation equations.
