Question

Find vectors u,v is the element of 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 $u=(u_1, u_2), v=(v_1, v_2).$

Given that $u$ is a scalar multiple of $(1, 3)$

$$u_1=k, u_2=3k$$

Given that $v$ is orthogonal  to $(1, 3)$

$$v_1+3v_2=0$$

Given that $u+v=(1, 2)$

$$u_1+v_1=1$$ $$u_2+v_2=2$$

We see that

$$u_2=3u_1$$ $$v_1=-3v_2$$

Substitute

$$u_1-3v_2=1$$ $$3u_1+v_2=2$$

Then

$$v_2=-3u_1+2$$ $$u_1+9u_1-6=1$$

$$u_1=0.7$$ $$v_2=-0.1$$ $$u_2=2.1$$ $$v_1=0.3$$

$u=(0.7, 2.1)$

$v=(0.3, -0.1)$