If , and are unit coplanar vectors, then the scalar triple product is
- A.0
- B.1
- C.
- D.
Show answer and solution
Correct option: A
Solution
Since a, b, c are coplanar vectors, [a b c] = 0.
The scalar triple product [2a−b, 2b−c, 2c−a] can be expanded using multilinearity. Writing it as a determinant of coefficients applied to [a b c]:
The transformation matrix from {a,b,c} to {2a−b, 2b−c, 2c−a} has coefficient matrix:
| 2 −1 0|
| 0 2 −1|
|−1 0 2|
So [2a−b, 2b−c, 2c−a] = det(M)·[a b c].
Since [a b c] = 0 (coplanar), the entire scalar triple product equals 0 regardless of det(M).
The scalar triple product [2a−b, 2b−c, 2c−a] can be expanded using multilinearity. Writing it as a determinant of coefficients applied to [a b c]:
The transformation matrix from {a,b,c} to {2a−b, 2b−c, 2c−a} has coefficient matrix:
| 2 −1 0|
| 0 2 −1|
|−1 0 2|
So [2a−b, 2b−c, 2c−a] = det(M)·[a b c].
Since [a b c] = 0 (coplanar), the entire scalar triple product equals 0 regardless of det(M).
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