MATH-205 Midterm: Bates MATH 205 100702johnson205examsolns
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1 p12 + 32 + ( 5)2 + 12 = 36 = 6 and p22 + ( 2)2 + 22 + 22 = 16 = 4. have lengths 6 and 4, respectively: and . Their dot product is 1 2+3( 2)+( 5)2+1 2 = 12, so the angle between them satis es cos = 12. Therefore = 120 , or = 2 . 2: we use elimination on an augmented matrix. First two steps: subtract three times the rst row from the second, and four times the rst row from the third. Next two steps: multiply the second row by 1, and then add twice the new second row to the third. Then the second equation becomes 11x2 + 85 = 63, so x2 = 2. Finally the rst equation becomes x1 10 + 30 = 22, so x1 = 2.
