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>

Expert Answers
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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.