# A1x + B1y + C1z + D1 = 0 and A2x + B2y + C2z + D2 = 0 are perpendicular. Find the value of A1A2 + B1B2 + C1C2.

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The planes A1*x + B1*y + C1*z + D1 = 0 and A2*x + B2*y + C2*z + D2 = 0 are perpendicular.

The normal to the plane A1*x + B1*y + C1*z + D1 = 0 is <A1, B1, C1> and the normal to A2*x + B2*y + C2*z + D2 = 0 is <A2, B2, C2>. If the two planes are perpendicular the dot product of the normals is 0.

`<A1, B1, C1> * <A2, B2, C2>` = `A1*A2 + B1*B2 + C1*C2` = 0

For the given planes the value of `A1*A2 + B1*B2 + C1*C2 = 0`

cosinusix | College Teacher | (Level 3) Assistant Educator

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The first plane is orthogonal to the vector `(A_1,B_1,C_1)`

The second plane is orthogonal to the vector `(A_2,B_2,C_2)` .

If the 2 planes are orthogonal, so are their orthogonal vectors.

Therefore `(A_1,B_1,C_1)` is orthogonal to `(A_2,B_2,C_2)`
and their dot product is 0.

`(A_1,B_1,C_1) .(A_2,B_2,C_2)=A_1A_2+B_1B_2+C_1C_2=0`

Solution: `A_1A_2+B_1B_2+C_1C_2=0`

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