Finding Point at Foot of Perpendicular of a plane from another point?

Question

Ok math people,
Plane 1: 3x+y-z=17
A point X is at the foot of the perpendicular from the point (2,1,6) to the Plane. Find the Coordinates of X.

How would I solve this? I don't necessarily need an answer, but I need to find some way of doing this.

Thanks a lot!

Answer 1

If you proceed from the given point along a vector perpendicular to the plane, you will intersect the plane at X. Therefore, find the line of all points along such a vector from the given point, and locate the point of intersection with the plane. You will need a vector perpendicular to the plane -- this is given by the coefficients of x, y, and z, so the vector is <3, 1, -1> (this always gives a perpendicular vector if the equation is in the form c₁x + c₂y + c₃z = k). The set of points found along this vector from (2, 1, 6) are (2+3t, 1+t, 6-t). Knowing that 3x+y-z=17, we substitute -- x=2+3t, y=1+t, z=6-t, so that:

3(2+3t) + (1+t) - (6-t) = 17

Expanding this:

11t + 1 = 17
11t = 16
t=16/11

Finally, the coordinates of x are (2+3t, 1+t, 6-t), which are (70/11, 27/11, 50/11).

Answer 2

Write the equation of the line thru A that is perpendicular to the plane and find the intersection of the line and the plane.

The equation of the plane is:

3x + y - z = 17

The point not on the plane is A(2,1,6).

The normal vector of the plane is also the directional vector of the perpendicular line.

The equation of the line is:

L = <2,1,6> + t<3,1,-1>
where t is a scalar that ranges over the real numbers

L = <2 + 3t, 1 + t, 6 - t>

Rewrite the equation in terms of t. This will give the value of t at the point of intersection with the plane.

3x + y - z = 17
3(2 + 3t) + 1(1 + t) - 1(6 - t) = 17
6 + 9t + 1 + t - 6 + t = 17
11t + 1 = 17
11t = 16
t = 16/11

Solve for X at t = 16/11.

L = <2 + 3t, 1 + t, 6 - t>

x = 2 + 3t = 2 + 3*(16/11) = 70/11
y = 1 + t = 1 + 16/11 = 27/11
z = 6 - t = 6 - 16/11 = 50/11

The point of intersection of the line and plane is

X(70/11, 27/11, 50/11)

Answer 3

Foot Of Perpendicular

Answer 4

good job bro.
U did it the simple and neat way, for once i actually understood how to do the sum.
Its good that u answered it from a Lehman's point of view (y)