Final answer:
The angle between the plane z=0 and the plane passing through the points (0,0,0), (3,2,0), and (0,2,1) can be found using the concept of the dot product and cross product. The angle is arccos(-6/sqrt(41)) radians.
Step-by-step explanation:
In mathematics, we can use the concept of the dot product and cross product to find the angle between two planes. First, find the normal vector to each plane. For the plane z=0, the normal vector is (0,0,1). For the second plane passing through the points (0,0,0), (3,2,0), and (0,2,1), we can get the normal vector by computing the cross product of the vectors formed by the points which is (2,-1,-6).
After finding the normal vectors, calculate the dot product of them, which is 0*2 + 0*-1 + 1*-6 = -6. At the same time, compute the magnitude of each normal, which are 1 and sqrt(2^2+1^2+6^2) = sqrt(41) for the first and second planes respectively. Thus, the cosine of the angle θ between the two planes is given by the dot product of the normal vectors divided by the product of their magnitudes, which yields cos(θ) = -6/sqrt(41). The actual angle θ can be obtained by taking the arccos of this quantity, which gives θ = arccos(-6/sqrt(41)) radians.
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