Question

If three vectors `bara, barb, barc` are not coplanar and `xbara + ybarb + zbarc = bar0`, then prove that x = y = z = 0

Answer 1

Suppose x ≠ 0

`xbara + ybarb + zbarc = bar0`     ...(Given)

∴ `xbara = -ybarb - zbarc`

∴  `bara = (-y/x)barb - (z/x)barc`    ...[∵ x ≠ 0]

∴ `bara` is a linear combination of `barb` and `barc`

∴  `bara, barb, barc` are coplanar

This is a contradiction

∴ x = 0

Similarly, y = z = 0

∴ x = y = z = 0