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Transition from the general equation of the line to the canonical equations
Consider the transition from the general equation of the straight line (10) to the canonical equations (11).
This transition is carried out according to ALGORITHM 1
Problem 16 Reduce the canonical form of the general equation of the line
Find the directing vector of the line. Since it should be perpendicular to normal vectors and given planes then can take a vector product of vectors and :
In this way,
As a point , through which the line passes, we can take the point of its intersection with any of the coordinate planes, for example, with the XOY plane, since then - and this point is determined from the system of equations of given planes, if we put in them :
Solving this system, we find: , , i.e.
Substitute the found coordinates of the point M 0 and the directing vector S into equation (2), we obtain
Do it yourself
Problem 16.1. Reduce the canonical form to the general equation of the line:
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