Given vectors $\overrightarrow{a}=(1,0,0)$, $\overrightarrow{b}=(1,1,0)$, and $\overrightarrow{c}=(1,1,1)$, find an orthonormal basis by using the Gram-Schmidt Method.

Step 1 ：Firstly, let's normalize the vector $\overrightarrow{a}$. The magnitude of $\overrightarrow{a}$ is $||\overrightarrow{a}||=\sqrt{1^2+0^2+0^2}=1$. So, the normalized vector $\overrightarrow{u_1}$ is $\overrightarrow{u_1}=\frac{1}{||\overrightarrow{a}||}\cdot \overrightarrow{a}=(1,0,0)$.

Step 2 ：Next, let's subtract the projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$ from $\overrightarrow{b}$ to get a vector orthogonal to $\overrightarrow{a}$. The projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$ is $\frac{\overrightarrow{b}\cdot \overrightarrow{a}}{||\overrightarrow{a}|{|}^2}\cdot \overrightarrow{a}=(1,0,0)$. So, the orthogonal vector $\overrightarrow{v}$ is $\overrightarrow{v}=\overrightarrow{b}-\frac{\overrightarrow{b}\cdot \overrightarrow{a}}{||\overrightarrow{a}|{|}^2}\cdot \overrightarrow{a}=(0,1,0)$. Then, normalize $\overrightarrow{v}$ to get $\overrightarrow{u_2}=\frac{1}{||\overrightarrow{v}||}\cdot \overrightarrow{v}=(0,1,0)$.

Step 3 ：Finally, subtract the projection of $\overrightarrow{c}$ onto $\overrightarrow{a}$ and $\overrightarrow{b}$ from $\overrightarrow{c}$ to get a vector orthogonal to $\overrightarrow{a}$ and $\overrightarrow{b}$. The projection of $\overrightarrow{c}$ onto $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\overrightarrow{c}\cdot \overrightarrow{a}}{||\overrightarrow{a}|{|}^2}\cdot \overrightarrow{a}+\frac{\overrightarrow{c}\cdot \overrightarrow{b}}{||\overrightarrow{b}|{|}^2}\cdot \overrightarrow{b}=(1,1,0)$. So, the orthogonal vector $\overrightarrow{w}$ is $\overrightarrow{w}=\overrightarrow{c}-\frac{\overrightarrow{c}\cdot \overrightarrow{a}}{||\overrightarrow{a}|{|}^2}\cdot \overrightarrow{a}-\frac{\overrightarrow{c}\cdot \overrightarrow{b}}{||\overrightarrow{b}|{|}^2}\cdot \overrightarrow{b}=(0,0,1)$. Then, normalize $\overrightarrow{w}$ to get $\overrightarrow{u_3}=\frac{1}{||\overrightarrow{w}||}\cdot \overrightarrow{w}=(0,0,1)$.