
Lines
A line is uniquely specified, when coordinates of 2 distinct points are known. In other words, only 1 line passes through 2 distinct points. Let and be those 2 points.
Let be any point on that line. Then,
since both the vectors are collinear. is any real number (or a scalar).
In other words,
, and are known as the direction ratios of the line.
If each of the direction ratios is divided by the distance between and , we get the direction cosines of the line. These are the cosines of the angles made by the line with , and axes respectively. Let them be and .
Having studied vectors, we know that the direction cosines of a line are the components of the unit vector along that direction. Hence,
(To verify the fact that these represent the cosines of the angles, we have to use the dot product of vectors. For example, consider . Unit vector along axis is . Taking the dot product we get )

Angle between any 2 lines is given by

Two lines are parallel if their direction cosines have a constant ratio. i.e.

2 lines are coincident if

2 lines are perpendicular if

Projection of a segment on a line
Let and be the endpoints of a segment and be direction cosines of a line . The projection of segment on line is given by

Planes
A plane is an entity formed by 3 noncollinear points. The general equation of plane is
are the direction ratios of the vector normal to the plane. This can be easily verified by using the fact that dot product of any 2 perpendicular vectors is 0.

If the plane passes through and has a normal with direction ratios , then the equation is

If are the intercepts made by the plane with $X,Y$ and $Z$ axes respectively, then
This is known as the slopeintercept form of the equation of plane.

If a plane is parallel to , its equation is

Angle between any 2 planes is the angle between their normals.

Length of perpendicular from a point on the plane is given by

Equation of a plane, passing through the intersection of 2 planes and is given by
is a parameter .

Equation of plane passing through and is given by
3 thoughts on “Three Dimensional Coordinate Geometry II”