Colinear

Colinear Colinear (or collinear) is a term used in geometry to describe points that lie on the same straight line. In a more general sense, colinearity can be applied to vectors, indicating that they are parallel and lie along the same line in a given vector space.

Definition

In Euclidean geometry, a set of points is said to be colinear if they all lie on a single straight line. For example, given three points A, B, and C, they are colinear if the slope of the line segment AB is the same as the slope of the line segment BC.

Properties

1. Linear Dependence: In linear algebra, vectors are colinear if one is a scalar multiple of the other. This implies that the vectors are linearly dependent. 2. Affine Geometry: In affine geometry, colinearity is preserved under affine transformations, which include translation, scaling, and rotation. 3. Coordinate Geometry: In coordinate geometry, points (x1, y1), (x2, y2), and (x3, y3) are colinear if the area of the triangle formed by these points is zero. This can be determined using the determinant of a matrix formed by these points.

Applications

Colinearity is a fundamental concept in various fields such as:

• Computer Graphics: Ensuring that points are colinear is crucial for rendering straight lines and edges.
• Robotics: Path planning often requires checking if multiple waypoints are colinear.
• Statistics: In regression analysis, colinearity (or multicollinearity) refers to the situation where two or more predictor variables are highly linearly related.

Related Concepts

• Concurrent lines
• Coplanar
• Linear independence
• Vector space

See Also

• Line (geometry)
• Point (geometry)
• Slope
• Affine transformation
• Determinant