A level revision

Vector geometry of planes

You need to be able to find the point of intersection between a line and a plane, find the angle between two planes and the distance between a plane and a poiint.

Intersection of a line and a plane

The intersection of a line and a plane is found first by creating equations in terms of for x, y and z from the equation of the line, substituting these into the equation of the plane to get a value of which can then be put back into the original line equation to find the equation of the point of intersection.

Example

Example question: Find the point between the line and the plane x + 2y + 3z = 11

Worked solution: Firstly write r as the general position vector:



From here we can create equations for x, y and z:







Substitute these into the plane equation:



Expand and simplify:



Putting back into the line equation:



Note that this is not the point of intersection, but the position vector of the point of intersection. The co-ordinates of the point of intersection is (0, 1, 3)

Angle between two planes

The angle between two planes is the same as the angle between their normal vectors, a and b using the equation:

To revise finding the angle between two vectors, see angles between vectors

Distance between a point and a plane

The distance from a point to a point on the place requires 3 steps, covered above. Firstly find the normal to the plane from the equation and hence find the equation of the line through the plane to the point. Then find the point of intersection between the line and the plane. Lastly find the distance between the point and the point of intersection.

Example

Example question: Find the distance between the point A(2, 3, 0) and the plane 2x + 5y + 3z = 0

Worked solution: Firstly, write the normal of the plane, which is just the coefficients of the x, y and z terms:



This gives the direction vector of the line from the plane to the pint A(2, 3, 0), thus the equation of this line is:



Next find the point of intersection of the line and the plane using the above technique. Write down equations for x, y, z:







Substitue these into the equation of the plane and solve for :





Put this back into the equation of the line to find the point of intersection:



Lastly the distance is found using pythagarus between the point of intersection and the original point, A(2, 3, 0):