Intersections and Distances

Intersections and Distances

Lines in Space

• In 3-dimensional space, a line can be represented as simultaneously satisfying a pair of parametric equations, often written in the form x = at + c, y = bt + d, z = et + f.
• The coefficients a, b, and e represent the direction ratios of the line, while c, d, and f are the coordinates of a specific point on the line.
• Any point on the line can be represented by a real number t, known as the parameter of the line.

Planes in Space

• Planes in 3-dimensional space can be represented by the coordinates satisfying a single linear equation of the form ax + by + cz = d.
• The coefficients a, b, and c are direction ratios of a vector n that is normal (perpendicular) to the plane, while d is a scalar quantity.
• Each point in the plane makes the same angle with the normal vector n.

Intersection of Lines and Planes

• A line intersects a plane if the parametric equations of the line can be substituted into the equation of the plane to produce a consistent set of equations.
• The intersection of two planes occurs along a line. The equation of this line can be found by equating the equations of the planes, and solving the resultant system.

Intersection of Lines

• Two lines intersect if their coordinates satisfy their respective parametric equations for a common value of the parameter t.
• Two lines are parallel if their direction ratios are proportional, and skew if they do not intersect yet are not parallel.

Distance Between Point and Line or Plane

• The perpendicular distance from a point to a line can be calculated using the cross product of the position vector of the point and the direction vector of the line, then dividing by the magnitude of the direction vector.
• The perpendicular distance from a point to a plane is calculated by substituting the coordinates of the point into the plane equation, and then dividing by the magnitude of the normal vector.

Distance Between Lines or Planes

• The distance between parallel lines or the distance between parallel planes can be found using a simple modification of the point-to-line or point-to-plane formula, where you calculate the perpendicular distance to any point on one of the lines or planes from the other.

Extenuating Situations

• If the lines or planes are not parallel, then they either intersect (in which case the distance is zero), or in the case of lines, may be skew lines, which do not intersect despite not being parallel. The shortest distance between skew lines can be found using vector methods.