Step 1:

$v_1=u_1=(1,1,1)$

Step 2:

$v_2=u_2-proj \ u_2$

= $ u_2-\frac {(u_2.v_1)}{||V_1||^2 }v_1$

$(u_2.v_1)=(-1,1,0)(1,1,1)$

$(u_2.v_1)=-1+1+0$

$(u_2.v_1)=0$

$||v_1||=\sqrt {1^2+1^2+1^2}=\sqrt{3}$

$||v_1||^2=3$

$v_2=(-1,1,0)-\frac{0}{3}(1,1,1)$

$v_2 = (-1,1,0)$

Step 3:

$v_3=u_3-proj u_3$

= $ u_3-\frac {(u_3.v_1)}{||V_1||^2 }v_1-\frac {(u_3.v_2)}{||V_2||^2 }v_2$

$(u_3.v_1)=(1,2,1)(1,1,1)$

$(u_3.v_1)=1+2+1$

$(u_3.v_1)=4$

$||v_1||=\sqrt {1^2+1^2+1^2}=\sqrt{3}$

$||v_1||^2=3$

$(u_3.v_2)=(1,2,1)(-1,1,0)$

$(-1+2+0)$

$(u_3.v_2)=1$

$||v_2||=\sqrt {-1^2+1^2+0^2}=\sqrt{2}$

$||v_2||^2=2$

$v_3=(1,2,1)-\frac{4}{3}(1,1,1)-\frac{1}{2}(-1,1,0)$

$v_3 = (\frac{1}{6},\frac{1}{6},\frac{-1}{3})$

$||v_1||=\sqrt {1^2+1^2+1^2}=\sqrt{3}$

$||v_2||=\sqrt {-1^2+1^2+0^2}=\sqrt{2}$

$||v_3||=\sqrt {\frac{1}{6}^2+\frac{1}{6}^2+\frac{-1}{3}^2}$

$||v_3||=\sqrt {\frac{1}{36}+\frac{1}{36}+\frac{1}{9}}$

=$\frac{\sqrt 6}{36}= \frac{1}{\sqrt 6}$

$q_1=\frac{v_1}{||v_1||}=\frac{1}{\sqrt3}(1,1,1)$

$q_1 = (\frac{1}{\sqrt3}, \frac{1}{\sqrt3}, \frac{1}{\sqrt3})$

$q_2=\frac{v_2}{||v_2||}=\frac{1}{\sqrt2}(-1,1,0)$

$q_2 = (\frac{-1}{\sqrt2}, \frac{1}{\sqrt2}, 0)$

$q_3=\frac{v_3}{||v_3||}=\frac{1}{\frac{1}{\sqrt6}}(\frac{1}{6},\frac{1}{6},-\frac{1}{3}$

$q_3={\sqrt6}(\frac{1}{6},\frac{1}{6},-\frac{1}{3})$

$q_3=(\frac{\sqrt6}{6},\frac{\sqrt6}{6},-\frac{\sqrt6}{3})$

$q_3 = (\frac{1}{\sqrt6}, \frac{1}{\sqrt6}, \frac{-2\sqrt6}{2 \times 3})$

$q_3 = (\frac{1}{\sqrt6}, \frac{1}{\sqrt6}, \frac{-2\sqrt6}{6})$

$q_3 = (\frac{1}{\sqrt6}, \frac{1}{\sqrt6}, \frac{-2}{\sqrt6})$