Vectors are physical quantities which like other quantities have a magnitude but also a direction linked to them. In the three-dimensional Cartesian system, position vectors are simply used to denote the location or position of the point, but a reference point is necessary.
Intercept form of the Equation of the Plane
There are infinite number of planes which are perpendicular to a particular vector as we have already discussed in our earlier sections. But when talking of a specific point only one exclusive plane occurs which is perpendicular to the point going through the given area. This can be denoted by this particular vector equation:
The denotation of this type of plane in a Cartesian equation is the following:
The direction ratios here are denoted by A, B, and C.
Also the equation of a plane crossing the three non-collinear points in vector form is given as:
The equation of a plane in Cartesian form passing through three non-collinear points is given as:
Let us now discuss the equation of a plane in intercept form.
The general equation of a plane is given as:
Ax + By + Cz + D = 0 (D ≠ 0)
Let us now try to determine the equation of a plane in terms of the intercepts which is formed by the given plane on the respective co-ordinate axes. Let us assume that the plane makes intercepts of a, b and c on the three co-ordinate axes respectively. Thus, the coordinates of the point of intersection of the plane with x, y and z axes are given by (a, 0, 0), (0, b, 0) and (0, 0, c) respectively.
Substituting these values in the general equation of a plane, we have
Aa + D = 0
Bb + D = 0
Cc + D = 0
From the above three equations, we have
Substituting these values of A, B, c and D in the general equation of the plane, we have
This gives us the required equation of a plane in the intercept form.
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