Asked by Hugsy
Very confused: The planes A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0 are perpendicular. Find the value of A1A2 + B1B2 + C1C2. Thanks!
Answers
Answered by
drwls
This one is a lot easier than it looks!
If the planes are perpendicular, the normals to the planes are perpendicular. The directions of the normals are given by the vectors
A1i + B1j + C1k
and
A2i + B2j + C2k
wjere i, j and k are mutually perpendicular unit vectors.
Now consider what the dot product of those two vectors is:
A1A2 + B1B2 + C1C2.
That dot product must be ZERO if the normals to the planes, and the plances themselves, are perpendicular.
If the planes are perpendicular, the normals to the planes are perpendicular. The directions of the normals are given by the vectors
A1i + B1j + C1k
and
A2i + B2j + C2k
wjere i, j and k are mutually perpendicular unit vectors.
Now consider what the dot product of those two vectors is:
A1A2 + B1B2 + C1C2.
That dot product must be ZERO if the normals to the planes, and the plances themselves, are perpendicular.
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