# What is the relationship of the given line L: (x,y,z) = (-2,1,5) + t<2,-8,-6> to each of the lines below? Is it coincident, parallel, intersecting or skew? 5. (-1,-3,2) + t<-3,12,9>

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### 1 Answer

You need to check if the given lines are coplanar, hence, you need to evaluate the following determinant, such that:

`d = [(-1+2,-3-1,2-5),(-3,12,9),(2,-8,-6)]`

`d = -72 - 72 - 72 + 72 + 72 + 54 => d = -18`

Since evaluating the determinant yields that `d = -18 != 0` , hence, the lines are not coplanar.

You need to check if the lines are parallel, hence, you need to check if the direction vectors of the lines are parallel such that:

`<2,-8,-6> = c*<-3,12,9> => {(-3c = 2),(12c = -8),(9c = -6):} => c = -2/3`

**Hence, since there exists solution to the system above, `c = -2/3` , thus, the direction vectors of the lines are parallel, hence, the given lines are parallel.**