Question

# Let u,v1 and v2 be vectors in R^3, and let c1 and c2 be scalars. If u is orthogonal to both v1 and v2, prove that u is orthogonal to the vector c1v1+c2v2.

Vectors and spaces

Let u,$$v_1$$ and $$v_2$$ be vectors in $$R^3$$, and let $$c_1$$ and $$c_2$$ be scalars. If u is orthogonal to both $$v_1$$ and $$v_2$$, prove that u is orthogonal to the vector $$c_1v_1+c_2v_2$$.

Since u is orthogonal to $$v_1$$ and $$v_2$$, we have that $$u\times v_1=0$$
$$u\times v_2=0$$
Now, $$u\times(c_1v_1+c_2v_2)=u\times(c_1v)+u(c_2v_2) =c_1u\times v_1+c_2u\times v_2 =c_1\times 0+c_2\times0 =0$$
Therefore, u is orthogonal to $$c_1v_1+c_2v_2$$, as required.