I thought I did these well, but my professor said there's an error with my number 3 and it is because distance - r can give a negative number and then this is what would be returned by 

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3. (15) Given a numpy array A1 which has shape (m,n) interpret the rows as points in n-dimensional Euclidean space, R". Let v be a 1 xn array, which will also be interpreted as a point in n-dimensional Euclidean space, and let r be a positive number. The hypersphere centered at v with radius r consists of all points (xo,..., n-1) such that r = (xovo) +...+(Xn-1 Vn-1), where vo, , Un-1 are the components of v. Create a function called closest_point that will take as input arrays A1, and v as described above and a positive number r, and returns the row index of A1 such that when the rows are interpreted as points in R", this row is the closest point of A1 to the the surface of the hypersphere centered at v with radius r. 4. (20) Given two numpy arrays A1 and A2 of shapes (m, n) and (m2, n) interpret the rows of the arrays as points of R. Define a function, called number_of_pairs, which will carry out the following steps. For each row of Al determine the row of A2 that is closest to it. Store the row index of the closest points in a array. Now carry out the same step but with the arrays reversed. Now return the number of pairs (a, a2) where a is a row of A1, a2 is a row of A2, the closest point of A2 to a is a2 and the closest point of Al to a2 is a. You may assume that for a given row there will only be one row in the corresponding array which is at the minimum distance. As a hint, once you have obtained arrays which give the indices of the closest points you may want to use one of these arrays as an index into the other array of indices. What is returned from this operation could be useful although a few additional operations may still be required. 3. (15) Given a numpy array A1 which has shape (m,n) interpret the rows as points in n-dimensional Euclidean space, R". Let v be a 1 xn array, which will also be interpreted as a point in n-dimensional Euclidean space, and let r be a positive number. The hypersphere centered at v with radius r consists of all points (xo,..., n-1) such that r = (xovo) +...+(Xn-1 Vn-1), where vo, , Un-1 are the components of v. Create a function called closest_point that will take as input arrays A1, and v as described above and a positive number r, and returns the row index of A1 such that when the rows are interpreted as points in R", this row is the closest point of A1 to the the surface of the hypersphere centered at v with radius r. 4. (20) Given two numpy arrays A1 and A2 of shapes (m, n) and (m2, n) interpret the rows of the arrays as points of R. Define a function, called number_of_pairs, which will carry out the following steps. For each row of Al determine the row of A2 that is closest to it. Store the row index of the closest points in a array. Now carry out the same step but with the arrays reversed. Now return the number of pairs (a, a2) where a is a row of A1, a2 is a row of A2, the closest point of A2 to a is a2 and the closest point of Al to a2 is a. You may assume that for a given row there will only be one row in the corresponding array which is at the minimum distance. As a hint, once you have obtained arrays which give the indices of the closest points you may want to use one of these arrays as an index into the other array of indices. What is returned from this operation could be useful although a few additional operations may still be required.