(Vector addition of points) Let A = (x1, y1) and let B = (x2, y2) be two points. Define
(Scalar multiplication of points) Let A = (x, y) and let r be a real number. Define
Let A and B be two points. The line segment between A and B is the set of points of the form
where
A and B are called the endpoints of the line segment between A and B.
The set of points of the form
where t > 1 are the points on the other side of B from A, and the set of such points where t < 0 are the points on the other side of A from B.
The set of points of the form
where t > 0 is the set of points on the line determined by A and B that are on the same side of A as B.
The set of points of the form
where t < 0 is the set of points on the line determined by A and B which are on the opposite side of A than B
The set of points of the form
where t > 0 is called the ray from A through B.
Given a line whose equation is
the set of points whose coordinates satisfy
is the set of points one side of the line, and the set of points whose coordinates satisfy
is the set of points on the other side of the line.
Let A, B, and C be three noncolinear points. The angle between AB and AC is the set of points which are on the same side of AB as C and the same side of AC as B. If a point is in the angle between AB and AC, it is said to be inside of the angle. Points which are not inside of the angle, and not on the rays starting at A and going through B and C, are said to be outside of the angle.
Let A, B, and C be three noncolinear points. The interior of the triangle ABC is the set of points which are on the same side of AB as C, the same side of AC as B, and the same side of BC as A. Points in the interior of the triangle are said to be inside of the triangle. Points which are not inside the triangle are said to be outside of the triangle.