Compute a·(b×c) for a = (1,2,3), b = (4,5,6), c = (7,8,9).

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Multiple Choice

Compute a·(b×c) for a = (1,2,3), b = (4,5,6), c = (7,8,9).

Explanation:
The scalar triple product a·(b×c) measures the signed volume of the parallelepiped formed by a, b, and c, and it is zero when the three vectors lie in the same plane (are coplanar). Compute b×c first: b×c = (5*9 − 6*8, 6*7 − 4*9, 4*8 − 5*7) = (−3, 6, −3). Now take the dot product with a: a·(b×c) = 1*(−3) + 2*6 + 3*(−3) = −3 + 12 − 9 = 0. So the result is 0, which also reflects that the three vectors are coplanar.

The scalar triple product a·(b×c) measures the signed volume of the parallelepiped formed by a, b, and c, and it is zero when the three vectors lie in the same plane (are coplanar).

Compute b×c first: b×c = (59 − 68, 67 − 49, 48 − 57) = (−3, 6, −3).

Now take the dot product with a: a·(b×c) = 1*(−3) + 26 + 3(−3) = −3 + 12 − 9 = 0.

So the result is 0, which also reflects that the three vectors are coplanar.

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