The equation of the line of intersection between two non parallel planes

Equation of the Line of Intersection Overview

To find the equation of the line of intersection between two non-parallel planes, one must use both the equations of the planes involved.
A line of intersection is a line that lies on both planes simultaneously. Each point on this line satisfies the equation of both planes.

Finding Line of Intersection Process

Start by recognising that the line of intersection will be perpendicular (or normal) to the normal vectors of both planes.
Calculate the cross product of the normal vectors of both planes using the determinant method. This gives the directional vector of the line.
Set the two plane equations equal to each other and solve to find a point that lies on this line of intersection. This gives the point on the line.
Use the vector equation of a line format, which is r = p + td, where r represents the position vector of any point on the line, p is a specific point on the line, t is a scalar and d is the directional vector.
The equation of the line of intersection is the point on the line plus t times the directional vector.

Special Considerations

Determining the equation of the line of intersection between two planes is an application of the vector form of the Cartesian equation of a line.
Ensure the planes are not parallel. If the planes are parallel, they do not intersect, and therefore, no line of intersection exists.
If the planes are identical, the line of intersection is the plane itself and therefore infinitely many solutions are possible.

Examples for Understanding

For example, to find the line of intersection between the planes x+y+z=3 and 2x-y+3z=7, the normal vectors are (1,1,1) and (2,-1,3). The cross product gives the directional vector (4,-5,1).
Solving the two planes’ equations simultaneously would provide a point on the line. Inserting this point and the directional vector into the vector equation of a line gives the equation of the line of intersection.

Potential Challenges

Understanding and applying the concepts of normal vectors and the cross product can be complex. Ensure to practise with a variety of questions.
Balance between simplifying your equations at each step and keeping track of all variables and vectors; a careful approach can prevent simple errors.
Remember to verify the conditions under which the result applies. Ensure the planes are indeed not parallel, as this impacts the validity of the line of intersection.

The equation of the line of intersection between two non-parallel planes is a fundamental topic in Further Pure 1’s module. Understanding and mastering this topic equips you with essential skills for three-dimensional geometry tasks.