Question

Find vectors u,v $\in$ R² such that u is a scalar multiple of (1,3), v is orthogonal to (1,3), and (1,2) = u +v.

Answer 1

Let $u=(u_1, u_2), v=(v_1, v_2).$

$u$ is a scalar multiple of $(1, 3):$ $u_2 =3u_1.$

$v$ is orthogonal to $(1,3):v_1+3v_2=0=>v_1=-3v_2$

$u+v=(1,2):u_1+v_1=1, u_2+v_2=2$

Hence

$$\begin{matrix} u_1-3v_2= 1\\ 3u_1+v_2 =2 \end{matrix}$$

$$\begin{matrix} u_1=3v_2+ 1\\ 9v_2+3+v_2 =2 \end{matrix}$$

$$\begin{matrix} u_1=0.7\\ v_2 =-0.1 \end{matrix}$$

Then

$$u=(0.7, 2.1),$$

$$v=(0.3, -0.1).$$