What is the relationship of the given line L : (x, y, z) = (1,-2,-3) + t<-4,-1,5> to each of the planes below?Plane contain L, plane is parallel to L or plane interests L?   4. -20x-20y-20z...

What is the relationship of the given line L : (x, y, z) = (1,-2,-3) + t<-4,-1,5> to each of the planes below?

Plane contain L, plane is parallel to L or plane interests L?

 

4. -20x-20y-20z = -4

Asked on by vaderrrrrr

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You need to test if the line L intersects or it is parallel to the given plane, hence, you need to substitute `1 - 4` `t` for x, `-2 -t ` for y and `-3 + 5t` for z in equation of plane, such that:

`-20(1 - 4t) - 20(-2 -t) - 20(-3 + 5t) = -4`

Dividing by -20 yields:

`1 - 4t - 2 - t - 3 + 5t = 1/5`

`-4 != 1/5`

Hence, the line does not intersect the given plane at any point.

You need to test if the line is parallel to the plane, hence, you need to evaluate the dot product of direction vector of the line and normal vector to the plane. If the dot product is zero, then the line is parallel to the plane.

`barL = <-4,-1,5>`

`bar n = <-20,-20,-20>`

You need to evaluate the dot product such that:

`<-4,-1,5>*<-20,-20,-20> = -4*(-20) + (-1)*(-20) + 5*(-20)`

`<-4,-1,5>*<-20,-20,-20> = 80 + 20 - 100 = 0`

Hence, evaluating the dot product of direction vector of the line and normal vector to the plane yields `barL*barn = 0` , thus the line L is parallel to the plane -`20x-20y-20z = -4.`

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