A Mathematical Blog

In this post, i will be writing about how to find the distance of a point from a plane from the coordinates of the point and the equation of the plane.  When we talk about the distance of a point to a plane we are talking about the shortest path connecting the point to the plane. The shortest path from a point to a plane is the path which is perpendicular to the plane. The length of the shortest path is what we are calling ‘the distance between the point and the plane’. If the point is ON the plane, then obviously the distance is zero. In the above picture, QR is the shortest path from point Q to plane P, where point R on the plane is the foot of the perpendicular.  Equation of P is 2x + y - 2z = 5. Let coordinates of the foot of the perpendicular be (a, b, c). Clearly, Q is not on P as the coordinates of Q does not satisfy the equation of P. So QR is a non-zero distance. (6 - 1 - 4 = 1 ≠ 0). Define a vector from Q to R(\(\overrightarrow{QR}\)) or from R to...

In two dimensional space, any two lines can only be either of the following : In tersecting or Parallel.  Similarly, in three dimensional space any given pair of planes can only be either intersecting or parallel.  The region of Intersection of any two intersecting planes is always a line. Consider two intersecting planes P1 & P2. Let letter L represent the line of intersection of the planes, as shown in the picture below.  Imagine a line L1 on plane P1 and a line L2 on plane P2 such that both these lines are perpendicular to the line of intersection L, as depicted below.  The angle between the planes is same as the angle between the lines (L1 & L2) perpendicular to the line of intersection, as seen in the picture above.  In the above picture, \(\theta\) is used to denote angle between lines L1 and L2(and planes). If \(\theta\) is the angle between intersecting lines, then 180 ° - \(\theta\) is also the angle between the lines, and hence a...