The Cartesian method for calculating in two dimensions begins by
establishing two reference directions, x and y, which are perpendicular to
each other, as sketched in the figure. The x arrow points in the "+"
direction. The opposite direction is called the "-" direction.
The + and - y directions are define by the y arrow in the same way. The x
arrow is called the "x axis" and the y arrow is called the "y
axis."A position vector which does not point in either the x or the y direction can be replaced by the sum of two vector components; one along the x direction and one along the y direction, as shown. The original vector is the vector sum of these two components, as sketched.
Because the x and y directions are perpendicular, the three vectors (original vector, x component and y component) form a right triangle. The magnitude (length) of the position vector is indicated either by vertical bars |R| or by omitting the arrow above the R. The Pythagorean theorem tells us that the magnitude of the position vector is related to the squares of the magnitudes of the components, as shown.
Trigonometry tells us the relations between the angle theta, the vector magnitude, and its components.
The components of a velocity vector are shown as a second example. In this case, the angle theta is different, but the form of the equations is the same. Note the (-) sign on the y component of the velocity vector. It indicates that the y component points in the -y direction (down in this example).