How to find a vector given the scalar triple productScalar triple product formula: a (...
How to find a vector given the scalar triple product
Scalar triple product formula:
If I know the value of the product and I know what vectors b and c are, how do I find what vector a is equal to?
Answer & Explanation
Unfortunately, you can't. Your real question is even simpler: if you know x and , can you solve for a?
The issue is that the dot product is not an invertible operation, so there are lots of ways to choose x such that . Think about it for the two-dimensional cases: let and . The hypothesis is that we know , , , and that
but this is a single equation with two unknowns ( and ), which has infinitely many solutions. If that seems odd, verify that and are both solutions