BE SPECIFC WITH ANSWERS: LIKE 1A, 1B, 1C, 2A, 2B, 3A, 3B......

1. Three computers, A, B, and C, have the numerical features listed below:

Feature	A	B	C
Processor Speed	3.06	2.68	2.92
Disk Size	500	320	640
Main-Memory Size	6	4	6

We may imagine these values as defining a vector for each computer; for instance, As vector is [3.06, 500, 6]. We can compute the cosine distance between any two of the vectors, but if we do not scale the components, then the disk size will dominate and make differences in the other components essentially invisible. Let us use 1 as the scale factor for processor speed, for the disk size, and for the main memory size. (a) In terms of and , compute the cosines of the angles between the vectors for each pair of the three computers. (b) What are the angles between the vectors if = = 1? (c) What are the angles between the vectors if = 0.01 and = 0.5?

2. A certain user has rated the three computers of Problem 1 as follows: A: 4 stars, B: 2 stars, C: 5 stars. (a) Normalize the ratings for this user. (b) Compute a user profile for the user, with components for processor speed, disk size, and main memory size, based on the data of Problem 1.

3. Given the following utility matrix, representing the ratings, on a 15 star scale, of eight items, a through h, by three users A, B, and C:

	a	b	c	d	e	f	g	h
A	4	5		5	1		3	2
B		3	4	3	1	2	1
C	2		1	3		4	5	3

Compute the following from the data of this matrix. (a) Normalize the matrix by subtracting from each nonblank entry the average value for its user. (b) Using the normalized matrix from Part (a), compute the cosine distance between each pair of users.