vector,distance?

Question

let say L:(1,3,2) + t (2,5,3) is parallel to the plane 2x+y+2Z=5

then what is the distance from the line to the plane??

is it same as the distance from point on the line L to the plane??

Answer 1

First let's see if the line is parallel to the plane. If the dot product of the directional vector to the line and the normal vector to the plane is zero, then they are parallel.

The equation of the line is:

L= <1,3,2> + t <2,5,3>

The directional vector v of the line is <2, 5, 3>.

The equation of the plane is:

2x + y + 2z = 5

The nomal vector n of the plane is <2, 1, 2>.

Take the dot product.

v • n = <2, 5, 3> • <2, 1, 2> = 4 + 5 + 6 = 15 ≠ 0

The dot product is not zero so the line and plane intersect. To find the point of intersection plug in the the coordinates for the line into the plane.

L= <1,3,2> + t <2,5,3> = <1 + 2t, 3 + 5t, 2 + 3t>

2x + y + 2z = 5
2(1 + 2t) + 1(3 + 5t) + 2(2 + 3t) = 5
2 + 4t + 3 + 5t + 4 + 6t = 5
15t + 9 = 5
15t = -4
t = -4/15

Solve for x, y, and z.

x = 1 + 2t = 1 + 2(-4/15) = 1 - 8/15 = 7/15
y = 3 + 5t = 3 + 5(-4/15) = 3 - 4/3 = 5/3
z = 2 + 3t = 2 + 3(-4/15) = 2 - 4/5 = 6/5

The point of intersection between line and plane is
(-7/15, 5/3, 6/5).

If however, the line had been parallel to the plane, all points on the line would have been the same distance from the plane. In that case you could find the distance from the line to the plane by picking any point on the line and finding its distance to the plane.

Answer 2

Yes it is the same, because they are parallel. You can take any point on the line L.