If A = (2,2-1) B = (3,1,2) find a non-zero vector C such that A*C = B*C = 0

To find a non-zero vector C such that A * C = B * C = 0, we can set up a system of equations using the dot product. Let's first calculate the dot product of A and C: $$A \cdot C = (2, 2, -1) \cdot (c_1, c_2, c_3) = 2c_1 + 2c_2 - c_3$$ Similarly, let's calculate the dot product of B and C: $$B \cdot C = (3, 1, 2) \cdot (c_1, c_2, c_3) = 3c_1 + 1c_2 + 2*c_3$$ Now we can set up the system of equations: $$2c_1 + 2c_2 - c_3 = 0$$ $$3c_1 + 1c_2 + 2*c_3 = 0$$ To solve this system of equations, we can use various methods such as substitution or elimination. Let's use the elimination method to solve it. Multiplying the first equation by -3 and the second equation by 2: $$-6c_1 -6c_2 + 3c_3 = 0$$ $$6c_1 + 2c_2 +4c_3 =0$$ Adding these two equations together: $$-4c_2 +7c_3=0$$ Now we have one equation with two variables. We can choose any value for one variable and solve for the other variable. Let's assume that C2 is equal to 1: $$-4\cdot1+7 c_3=0 $$$$c_3=\frac{4}{7}$$ Therefore, a non-zero vector C that satisfies $$A \cdot C = B \cdot C = 0$$ $$is:\: C=(c_1,c_2,c_3)=(-\frac{5}{7},-\frac{4}{7},1)$$