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!
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.
Well, if the planes A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0 are perpendicular, then the dot product of their normal vectors should be zero.
The normal vector of the first plane is (A1, B1, C1), and the normal vector of the second plane is (A2, B2, C2).
So, to find the dot product, we multiply the corresponding components and sum them up, giving us:
A1A2 + B1B2 + C1C2.
Therefore, the value of A1A2 + B1B2 + C1C2 is the key to finding out if those planes are perpendicular.
To find the value of A1A2 + B1B2 + C1C2, we can use the concept of perpendicular vectors. Two vectors are perpendicular if their dot product is zero.
The given equations of the planes can be written as:
Plane 1: A1x + B1y + C1z + D1 = 0
Plane 2: A2x + B2y + C2z + D2 = 0
Consider the normal vectors of the two planes. The coefficients of x, y, and z in the equation of each plane represent the components of its normal vector.
For Plane 1, the normal vector is (A1, B1, C1).
For Plane 2, the normal vector is (A2, B2, C2).
Since the two planes are perpendicular, their normal vectors are also perpendicular. This means their dot product is zero:
(A1, B1, C1) · (A2, B2, C2) = 0
Expanding the dot product, we have:
A1A2 + B1B2 + C1C2 = 0
Therefore, the value of A1A2 + B1B2 + C1C2 is zero.
To find the value of A1A2 + B1B2 + C1C2, we need to use the fact that the two planes A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0 are perpendicular.
When two planes are perpendicular, the dot product of their normal vectors is zero. The normal vector of a plane is a vector perpendicular to the plane and can be obtained from the coefficients of the variables in the plane equation.
Let's start by finding the normal vectors of the two planes. The normal vector of A1x + B1y + C1z + D1 = 0 is (A1, B1, C1), and the normal vector of A2x + B2y + C2z + D2 = 0 is (A2, B2, C2).
Now, we can calculate the dot product of the two normal vectors:
(A1, B1, C1) · (A2, B2, C2) = A1A2 + B1B2 + C1C2
Since the two planes are perpendicular, their normal vectors are perpendicular, and their dot product is zero. Therefore, A1A2 + B1B2 + C1C2 = 0.
So, the value of A1A2 + B1B2 + C1C2 is zero.
Note: This method works because the dot product of two perpendicular vectors is zero. If the planes were not specified as perpendicular, we would need additional information to find the value of A1A2 + B1B2 + C1C2.