Question 1035834
z is orthogonal to x iff *[tex \large z \cdot x = 0]. Similarly with z and y.


Let *[tex \large x = (x_1, x_2, x_3, x_4)] and *[tex \large y = (y_1, y_2, y_3, y_4]. Hence we wish to find a vector *[tex \large z = (z_1, z_2, z_3, z_4)] such that both are true:


*[tex \large x_1z_1 + x_2z_2 + x_3z_3 + x_4z_4 = 0]
*[tex \large y_1z_1 + y_2z_2 + y_3z_3 + y_4z_4 = 0]

This is a system of two equations with four variables, and one can find a nonzero solution z that works (this follows from the rank-nullity theorem in linear algebra, since the matrix whose entries are x_1, ..., x_4 \\ y_1, ..., y_4 has non-zero nullity, so the nullspace contains some nonzero vector which we can choose for z).