1) Find an equation of the plane consisting of all points that are equidistant f

Question

1) Find an equation of the plane consisting of all points that are equidistant from (3, 3, 2) and (-1, -4, 5), and having -4 as the coefficient of x .

2) Find the distance from the point (4,4,5) to the line x=0, y=4+5t, z=5+4t

3) Find an equation of a plane containing the three points (-1, 5, 1), (-3, 10, 6), (-3, 11, 8)  in which the coefficient of x is 5.

Your help would be much appreciated!

Find an equation of the plane consisting of all points that are equidistant from (3, 3, 2) and (-1, -4, 5), and having -4 as the coefficient of x . Find the distance from the point (4,4,5) to the line x=0, y=4+5t, z=5+4t Find an equation of a plane containing the three points (-1, 5, 1), (-3, 10, 6), (-3, 11, 8) in which the coefficient of x is 5.

Explanation / Answer

2) Let u = vector from point ( 4, 4,5) to some point on the line such as (0, 4, 5)

u = < -4, 0, 0 >

Let v = direction vector of line: v = < 0, 5, 4 >

u x v = < -4, 0, 0 > x < 0, 5, 4 >
u x v = < 0, 16, -20 >

Shortest distance from point (4,4,5) to line:
d = | u x v | / | v |
d = ?(0+16^2+20^2) / ?(0+25+16) = ?656 / ?41= 4
d = 4

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