Question

For any three vectors −→a , −→b , and −→c , we always have (−→a × −→b ) × −→c = −→a × ( −→b × −→c ).

a)true
b)false

Answer 1

When two vectors, represented by −→a and −→b, are cross-multiplied, the resulting vector is perpendicular to the plane defined by the original vectors. So, the statement (−→a × −→b ) × −→c = −→a × ( −→b × −→c ) is true.

Step-by-step explanation:

The equality (−→a × −→b) × −→c = −→a × (−→b × −→c) is indeed valid.

This is grounded in the geometric properties of vector cross products.

When two vectors, represented by −→a and −→b, are cross-multiplied, the resulting vector is perpendicular to the plane defined by the original vectors.

Taking this resultant vector and cross-multiplying it with a third vector, −→c, once again produces a vector perpendicular to the plane formed by the two vectors −→a and −→b.

This geometric insight illustrates that the order of multiplication in the expression does not influence the outcome.

Consequently, the specified order of operations remains interchangeable without altering the final vector result.

In essence, the equality holds true because each cross product operation consistently yields a vector that is orthogonal to the planes spanned by the involved vectors.