Vector equation of straight lines and applicaitons

A straight line in three-dimensional space can be represented by a vector equation, which is an equation that defines the position of a point on the line in terms of a direction vector and a point on the line. The general form of the vector equation of a straight line is given by:

P = P0 + tD

where P is a point on the line, P0 is a known point on the line, t is a scalar parameter, and D is a direction vector that defines the direction of the line. The scalar parameter t can take any value, and the line can be parameterized by varying t from negative infinity to positive infinity.

For example, if P0 = (x0, y0, z0) and D = (a, b, c), then the equation of the line can be written as:

x = x0 + ta y = y0 + tb z = z0 + tc

This equation describes a straight line passing through the point P0 and having the direction vector D.