 # Coplanarity of Two Lines

• Two lines \frac{x-x_{1}}{a_{1}}=\frac{y-y_{1}}{b_{1}}=\frac{z-z_{1}}{c_{1}} and \frac{x-x_{2}}{a_{2}}=\frac{y-y_{2}}{b_{2}}=\frac{z-z_{2}}{c_{2}} are coplanar if \begin{vmatrix}x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1}\\ a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\end{vmatrix} = 0
• Two lines \vec{r}=\vec{a}_{1}+\lambda \vec{b}_{1} and \vec{r}=\vec{a}_{2}+\mu \vec{b}_{2} are intersected if (\vec{a}_{2}-\vec{a}_{1})\cdot (\vec{b}_{1}\times \vec{b}_{2}) = 0
• Two lines \frac{x-x_{1}}{a_{1}}=\frac{y-y_{1}}{b_{1}}=\frac{z-z_{1}}{c_{1}} and \frac{x-x_{2}}{a_{2}}=\frac{y-y_{2}}{b_{2}}=\frac{z-z_{2}}{c_{2}} are intersected if \begin{vmatrix}x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1}\\ a_{1} & b_{1} & c_{1}\\ a_{2} & b_{2} & c_{2}\end{vmatrix} = 0

### Part2: View the Topic in this video From 01:00 To 06:33

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1. Two planes \vec{r}= \vec{a}_{1}+\lambda \vec{b}_{1} and \vec{r}= \vec{a}_{2}+\mu \vec{b}_{2} are coplanar if (\vec{a}_{2}-\vec{a}_{1})\cdot(\vec{b}_{1}\times\vec{b}_{2})=0

2. Two planes a1x + b1y + c1z + d1 = 0 and a2x + b2y + c2z + d2 = 0 are coplanar if \begin{vmatrix}x_{2}-x_{1} & y_{2}-y_{1} & z_{2}-z_{1} \\a_{1} & b_{1} & c_{1} \\a_{2} & b_{2} & c_{2} \end{vmatrix}=0.