# Linear Algebra

posted by .

In 3-space, a plane abc is spanned by three points a, b, c. The point x does not lie on this plane.

I'm trying to find the formulas used to find the point p on abc with minimal distance to x.

I know the point p must be perpendicular to the point p and the plane.

So, I began with taking a vector from the plane to the point, w:

x = (x0,x1,x2)
p = (p0,p1,p2) <-- unknown

w = [p0-x0, p1-x1, p2-x2]

v = [a, b, c] = [a0 b0, c0]
[a1, b1, c1]
[a2, b2, c2]

then I project w onto v

D = |(v dot w)|/|v|

but my problem arises here because I do not know the distance and I do not know the coordinates of p. Am I supposed to be using the gradient formula instead?

Any help is greatly appreciated! Thank you

• Linear Algebra -

First shift everything by minus a, so that the plane goes through the origin. The plane then goes through the points:

b'= b - a

and

c'= c - a

and through the origin.

Then, we define an orhonormal basis for the linear space spanned by the plane. We take the first basis vector to be in the direction of b'. So we normalize b1 to obtain:

e1 = b'/|b'|

From c' we subtract the component in the direction of e1:

f2 = c' - (c' dot e1) e1

Then f2 is orthogonal to e1:

f2 dot e1 = c'dot e1 -
(c' dot e1) (e1 dot e1.

Now, e1 dot e1 = 1, because e1 is normalized. So we see that
f2 dot e1 =0.

Then we have to normalize f2 to get the second basis vector:

e2 = f2/|f2|

You can now simply project the point x'= x-a onto the plane to obtain p':

p' = (x' dot e1) e1 + (x' dot e2) e2

Then shift back by a to obtain p:

p = p'+ a

## Similar Questions

1. ### Maths - Vectors

The position vectors a,b,c of the points A,B,C relative to the origin O are ai, bj and ck respectively.Find: 1) a vector equation for the line l throught O which is normal to the plane ABC. 2) a vector equation of the plane ABC The …
2. ### maths - geometry

In a space with an orthonormal coordinate system consider the plane; &: 4x-3y=12 (a)(i)Find the coordinates of the points of intersection of the plane & with the coordinates axes (=axes intersections?
3. ### Algebra

Let A(4,3) B(5,8) and C(3,10) be three points in a coordinate plane. Find the coordinates of a point D such that the points ABC and D form a parallelogram with (a)AB as one of the diagonals (b)AC as one of the diagonals (c)BC as one …
4. ### Precalc

At the point were a line intersects a plane [with the equation (24x+32y+40z=480) and three points of A(5, 10, 1), B(6, 3, 6), and C(12, 1, 4)], the vector i-, vector j-, and vector k-coefficients of the line equal the corresponding …
5. ### Geometry

4.Determine which point does not lie on the graph of the line y=x-3 a. (-10, 13) b. (-10-13) c. (-8, -11) A?
6. ### math

Point A and B of a triangle ABC are at (-1,1) and (3,9), respectively; while point C is on the parabola y=x2. Find coordinate of C so that the area of ABC is minimum, and calculate the largest are of ABC.
7. ### maths

Let ABC be a triangle in the plane. Find circles C0;C1; : : : ;C6 such that Cj has exactly j points in common with the boundary of ABC (this \boundary" consists of the line segments AB, BC, CA). Is it possible to find a circle C7 with …
8. ### Linear Algebra

1. Considering 2 planes with equations: x + 2y - 3z = 1 x + 2y - 3z = 18 (a) Given (a,b,c) a point on the first plane, find the point where the line perpendicular to both planes passes by (a,b,c) and through the 2nd plane.
9. ### maths

Let ABC be a triangle in the plane. Find circles C0;C1; : : : ;C6 such that Cj has exactly j points in common with the boundary of ABC (this \boundary" consists of the line segments AB, BC, CA). Is it possible to nd a circle C7 with …
10. ### Geometry

Plane R contains points ,M,and Y. Plane Z intersects with plane R at SM, AY intersects plane R at point Y. . Point A is not in plane Z or plane R. MY forms a right angle with MS. A. What is measure of < SMY?

More Similar Questions