1. Using the process of computing the normal equation to a plane in Problem 6.23, find an equation for a plane involving a determinant Problem 6.23 The equation of a plane is determined by three...

1. Using the process of computing the normal equation to a plane in Problem 6.23, find an equation for a plane involving a determinant

Problem 6.23

The equation of a plane is determined by three non-collinear points and Take the two vectors
 and
in the plane and use the cross product to find a vector n perpendicular to the plane. Pick an arbitrary point
 in the plane and require that
 This generates what is called the normal equation of a plane.

 Draw a figure that illustrates this process.

 Find the equation of the plane containing the points

2. Show that if
 is an orthogonal matrix, and
 and
 are vectors in

ⁿ
 then
.

3. Show that if
 and
 are orthogonal matrices, then
 and
 are orthogonal matrices.

4. The matrix A has orthogonal columns. Convert it to an orthogonal matrix by normalizing the columns.