Instruction

1

Let the plane specified by three points belonging to it K(xk;yk;zk), M(xm;ym;zm), P(xp;yp;zp). To find the normal vector to make the equation of this

**plane**. Mark an arbitrary point lying on**the plane**, with the letter L, I want her to have coordinates (x;y;z). Now consider the three vectors PK, PM and PL, they lie on the same**plane**(coplanar), so their scalar triple product is zero.2

Find the coordinates of vectors PK, PM, and PL:

PK = (xk-xp, yk-yp;zk-zp)

PM = (xm-xp, ym-yp;zm-zp)

PL = (x-xp, y-yp z-zp)

Mixed product of these vectors is equal to the determinant presented in the figure. This determinant should be calculated to find the equation for

PK = (xk-xp, yk-yp;zk-zp)

PM = (xm-xp, ym-yp;zm-zp)

PL = (x-xp, y-yp z-zp)

Mixed product of these vectors is equal to the determinant presented in the figure. This determinant should be calculated to find the equation for

**the plane**. The calculation of the mixed product for a particular case, see example.3

Example

Let a plane defined by three points K(2;1;-2), M(0;0;-1) and P(1;8;1). You want to find a normal vector

Take an arbitrary point L with coordinates (x;y;z). Compute the vectors PK, PM, and PL:

PK = (2-1;1-8;-2-1) = (1;-7;-3)

PM = (0-1;0-8;-1-1) = (-1;-8;-2)

PL = (x-1;y-8;z-1)

Make the determinant for the scalar triple product of vectors (it is on the picture).

Let a plane defined by three points K(2;1;-2), M(0;0;-1) and P(1;8;1). You want to find a normal vector

**to the plane**.Take an arbitrary point L with coordinates (x;y;z). Compute the vectors PK, PM, and PL:

PK = (2-1;1-8;-2-1) = (1;-7;-3)

PM = (0-1;0-8;-1-1) = (-1;-8;-2)

PL = (x-1;y-8;z-1)

Make the determinant for the scalar triple product of vectors (it is on the picture).

4

Now decompose the determinant along the first row, and then count the value of the determinant of size 2 by 2.

Thus, the equation

Thus, the equation

**of plane**is 10x + 5y - 15z - 15 = 0 or what is the same, -2x + y - 3z - 3 = 0. It is easy to determine the vector normal to**plane**: n = (-2;1;-3).