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: a ( b × c )
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




2022-07-21Added 14 answers

Unfortunately, you can't. Your real question is even simpler: if you know x and a x = y, 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 a x = y. Think about it for the two-dimensional cases: let a = [ a 1 , a 2 ] and x = [ x 1 , x 2 ]. The hypothesis is that we know x 1 , x 2 , y, and that
a 1 x 1 + a 2 x 2 = y ,
but this is a single equation with two unknowns ( a 1 and a 2 ), which has infinitely many solutions. If that seems odd, verify that a = [ 0 , y x 2 ] and a = [ y x 1 , 0 ] are both solutions

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