# 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? 1. (1, -3, 1) + t<1,2,3>

sciencesolve | Teacher | (Level 3) Educator Emeritus

Posted on

You need to check if the lines are parallel, hence, you need to test if the direction vectors are scalar multiples of each other, such that:

`<2,-8,-6> = k*<1,2,3> => <2,-8,-6> = <k,2k,3k>`

`{(k = 2),(2k = -8),(3k = -6):}=> {(k = 2),(k = -4),(k = -2):}`

Since the values of k does not coincide, the lines are not parallel.

You need to test if the lines intersect, hence, you need to equate x,y and z coordinates, such that:

`2t_1 - 2 = t_2 + 1 => t_2 = 2t_1 - 3`

`-8t_1 + 1 = 2t_2 - 3 => -8t_1 + 1 = 4t_1 - 6 - 3 => -12t_1 = -10` `=> t_1 = 5/6`
`=> t_2 = 5/3 - 3 = -4/3`

You need to substitute `5/6` for `t_1` and `-4/3 ` for `t_2` in equation `-6t_1 + 5 = 3t_2 + 1` such that:

`-5 + 5 = -4 + 1 => 0 != -3 =>`   the lines does not intersect

Hence, since the lines are neither parallel, nor intersect each other, thus, the lines are skew.