Answer key for exam #1 are 12 + 12 + 12 + 02 + 12 = 4 = 2 and 12 + 12 + 12 + 22 + 12 = The dot product of the vectors is 1 + 1 + 1 + 0 + 1 = 4, so the angle between them satis es cos = 4 , or 45 : we use elimination on an augmented matrix. First two steps: subtract the rst row from the second, and three times the rst row from the third. Next two steps: multiply the second row by 1, and then add twice the second row to the third. Then the second equation becomes 2x2 6 = 8 or x2 3 = 4, so x2 = 7. Finally the rst equation becomes x1 + 28 + 10 = 3, so x1 = 35. 3(a) step 1: subtract twice the rst row of a from the others.
