Given any three non-zero vectors A, B and C, their triple product Ā.(BxC) vanishes if
(a) They are perpendicular to each other
b) Any two of them are perpendicular
c) Any two of them are parallel
d) they are non coplaner?

Question

Answer

Perpendicular Vectors
Perpendicular vectors are vectors that form a right angle with each other. If vectors A, B, and C are perpendicular to each other, then their triple product Ā.(BxC) will be zero. This is because the cross product of B and C will be perpendicular to both B and C, and when you take the dot product of A with this perpendicular vector, the result will be zero.

Two Perpendicular Vectors
If any two of the vectors A, B, and C are perpendicular to each other, their triple product will also be zero. This is because the cross product of the two perpendicular vectors will be perpendicular to both of them, resulting in a dot product of zero when taking the dot product with the third vector.

Parallel Vectors
If any two of the vectors A, B, and C are parallel to each other, their triple product will not be zero. This is because the cross product of two parallel vectors will be zero, resulting in a non-zero dot product when taking the dot product with the third vector.

Non-coplanar Vectors
If the vectors A, B, and C are non-coplanar, their triple product will not be zero. This is because in order for the triple product to be zero, the vectors must lie in the same plane, which is not the case for non-coplanar vectors.

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