How to Find a Line in Parametric Form That Is Perpendicular to the Plane

Finding a line in parametric form that is perpendicular to a given plane is a vector calculus problem that requires an understanding of certain concepts covered in prerequisite math classes. However, though understanding the thinking behind these problems is difficult, the actual mechanics of solving the problem are almost deceptively easy. One thing that often troubles students and teachers alike is that many of these problems have infinitely many solutions, since the line in question could be at any location on the plane without affecting its perpendicularity.

Instructions

    • 1

      Rearrange the plane equation until it is in the standard form ax + by + cz = d, where a, b, c and d are constants. Note that not all plane equations use all three variables; if it makes it easier for you to visualize, put a coefficient of 0 in front of the missing variables, as in 0x + 2y + 3z = 4.

    • 2

      Find the normal vector to the plane. The coefficients of the vectors are the same as the coefficients in the plane equation, giving you ai + bj + ck as the normal vector. The constant d makes no difference, since it does not affect the angle of the plane.

    • 3

      Convert to parametric form. If there is a point through which the line must pass, these coordinates serve as the constants, and the coefficients from the normal vector are the coefficients of t regardless. For example, a line perpendicular to x + 2y + 3z = 4 and passing through (5, 6, 7) would have parametric equations x = t + 5, y = 2t + 6, and z = 3t + 7.

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