Final answer:
To find the angle between the surfaces at the point (2, -1, 2), we determine the normal vectors to each surface at that point and use the dot product formula to calculate the angle, which is approximately 51.87 degrees.
Step-by-step explanation:
The student is asking to find the angle between the surfaces represented by the equations x² + y² + z² = 9 and x² + y² - z = 3 at the point (2, -1, 2). To do this, we need to find the normal vectors to these surfaces at the given point and then calculate the angle between these normals.
For the first surface, we can find the normal vector by taking the gradient of the function, which gives us (2x, 2y, 2z) at any point on the surface. At the given point, the normal vector A is (4, -2, 4).
Similarly, for the second surface, the gradient gives us the normal vector (2x, 2y, -1) at any point. At the given point, the normal vector B is (4, -2, -1).
The angle between the normal vectors can be found using the dot product formula: A · B = |A||B|cos(θ), where θ is the angle between the vectors.
The dot product of vectors A and B is calculated as:
4*4 + (-2)*(-2) + 4*(-1) = 16 + 4 - 4 = 16
The magnitudes of vectors A and B are:
|A| = √(4² + (-2)² + 4²) = √(16 + 4 + 16) = √36 = 6
|B| = √(4² + (-2)² + (-1)²) = √(16 + 4 + 1) = √21
Substituting these into the dot product formula and solving for θ gives:
cos(θ) = (16)/(6*√21)
θ = cos⁻¹(16/(6*√21)) ≈ cos⁻¹(0.619)
Computing this, we get the angle θ is approximately 51.87 degrees.