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Let a = (1, 2, -1), b = (2, -1, 3) and c = (t, 1, 2), where t is a…
Question
Let a = (1, 2, -1), b = (2, -1, 3) and c = (t, 1, 2), where t is a real parameter. The parallelepiped spanned by a, b, c has volume 21. Find the sum of all real values of t for which this holds.
✓ Verified answer: 6checked by our engine — not a guess
Step-by-step solution
The volume of the parallelepiped equals the absolute value of the scalar triple product [a b c] = det[a b c].
Expanding the determinant with columns a, b, c gives [a b c] = 5t - 15.
Setting |5t - 15| = 21 gives two cases: 5t - 15 = 21 → t = 36/5, and 5t - 15 = -21 → t = -6/5.
Sum = 36/5 + (-6/5) = 30/5 = 6.
Final answer6
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