Maths - Projections of lines on planes

(A B)x = ((Az * Bx - Bz * Ax) * Bz - By * (Ax * By - Bx * Ay)) / (Bx² + By² + Bz²)
(A B)y = ((Ax * By - Bx * Ay) * Bx - Bz * (Ay * Bz - By * Az)) / (Bx² + By² + Bz²)
(A B)z = ((Ay * Bz - By * Az) * By - Bx * (Az * Bx - Bz * Ax)) / (Bx² + By² + Bz²)

grouping terms,

(A B)x = (Az * Bx* Bz - Bz * Ax* Bz - Ax * By* By + Bx * Ay* By) / (Bx² + By² + Bz²)
(A B)y = (Ax * By* Bx - Bx * Ay* Bx - Ay * Bz*Bz + By * Az*Bz) / (Bx² + By² + Bz²)
(A B)z = (Ay * Bz* By - By * Az* By - Az * Bx*Bx + Bz * Ax*Bx) / (Bx² + By² + Bz²)

In matrix form,

(A B) = 1 / (Bx2 + By2 + Bz2)*

-Bz* Bz- By* By Bx * By Bx* Bz By* Bx -Bx*Bx- Bz*Bz By *Bz Bz *Bx Bz* By -By*By- Bx*Bx

[A]

I think the signs of all terms should be inverted, see error heading below, can anyone help?

Alternative Form As pointed out here, If B is normalised (unit length) then Bx2 + By2 + Bz2 =1 so we get: (A B) = Bx*Bx - 1 Bx * By Bx* Bz By* Bx By*By- 1 By *Bz Bz *Bx Bz* By Bz*Bz - 1 [A] which is: (A B) = ( Bx By Bz * Bx By Bz -[I])[A] which is: (A B) = (B * Bt - [I]) [A] where: [I] = unit matrix Bt = transpose of B vector As this is quite a simple equation I wonder if there is a simpler way to derive it? perhaps the method of least squares discussed here.

parallel component

A || P = A • P * P/|B|²

A • P can be calculated as follows:

Ax * Bx + Ay * By + Az * Bz

so,

(A || P)x = (Ax * Bx + Ay * By + Az * Bz) * Bx / (Bx² + By² + Bz²)
(A || P)y = (Ax * Bx + Ay * By + Az * Bz) * By/ (Bx² + By² + Bz²)
(A || P)z = (Ax * Bx + Ay * By + Az * Bz) * Bz/ (Bx² + By² + Bz²)

in matrix form this is:

(A || P) = 1 / (Bx2 + By2 + Bz2)*

Bx2 Bx * By Bx * Bz By * Bx By2 By * Bz Bz * Bx Bz * By Bz2

[A]

Alternative Form If B is normalised (unit length) then Bx2 + By2 + Bz2 =1 so we get: (A || P) = Bx * Bx Bx * By Bx * Bz By * Bx By * By By * Bz Bz * Bx Bz * By Bz * Bz [A] which is: (A P) = ( Bx By Bz * Bx By Bz )[A] which is: (A P) = (B * Bt) [A] where: Bt = transpose of B vector As this is quite a simple equation I wonder if there is a simpler way to derive it? perhaps the method of least squares discussed here.

Error

we have shown above that:

A = A || B + A B

which would give:

A = (B * B^t - [I] + B * B^t)[A]

which should simplify to A = [I][A] but it does not so it looks like the sign of (A B) is inverted can anyone see where I went wrong?