For the following exercises, lines $L_1$ and $L_2$ are given.Determine whether the lines are equal, parallel but not equal, skew, or intersecting.$L_1:x=2t,y=0,z=3,t\in \mathbb{R}$ and$L_2:x=0,y=8+s,z=7+s,s\in \mathbb{R}$

Step by Step Solution: Step 1. To determine if $L_1$ and $L_2$ are parallel or intersecting, we will check whether their direction vectors are parallel or not. Step 2. The direction vector of $L_1$ is $\overrightarrow{v_1}=\left[\begin{array}{c}2\\ 0\\ 0\end{array}\right]$ and the direction vector of $L_2$ is $\overrightarrow{v_2}=\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]$. Step 3. Since $\overrightarrow{v_1}$ and $\overrightarrow{v_2}$ are not parallel, $L_1$ and $L_2$ are not parallel. Step 4. Next, we will check if $L_1$ and $L_2$ intersect. To do this, we will find the point on each line that has the same $x$, $y$, and $z$ value since the point of intersection must satisfy both equations. Step 5. For $L_1$, the $x$-coordinate is $2t$, the $y$-coordinate is $0$, and the $z$-coordinate is $3$. Step 6. For $L_2$, the $x$-coordinate is $0$, the $y$-coordinate is $8+s$, and the $z$-coordinate is $7+s$. Step 7. Setting $2t=0$, we get $t=0$. Substituting $t=0$ into the parametric equation of $L_1$, we get the point $P_1=(0,0,3)$. Setting $8+s=0$ and $7+s=3$, we get $s=-8$ and $s=-4$, respectively. Hence, the point on $L_2$ with the same $x$, $y$, and $z$ values is $P_2=(0,-8,-1)$. Step 8. Since $P_1\ne P_2$, $L_1$ and $L_2$ are not equal. Similarly, since $\overrightarrow{v_1}$ and $\overrightarrow{v_2}$ are not orthogonal, $L_1$ and $L_2$ are not skew. Step 9. Therefore, $L_1$ and $L_2$ are intersecting. Final Answer: The lines $L_1:x=2t,y=0,z=3,t\in \mathbb{R}$ and $L_2:x=0,y=8+s,z=7+s,s\in \mathbb{R}$ are intersecting.