More on Matrices - 3D Math Primer for Graphics and Game Development - page 173

This gives us nine equations, all of which must be true for M to be orthog-
onal:

m 11 m 11 + m 12 m 12 + m 13 m 13 = 1, (6.8)

Conditions satisfied by
an orthogonal matrix

m 11 m 21 + m 12 m 22 + m 13 m 23 = 0,
m 11 m 31 + m 12 m 32 + m 13 m 33 = 0,
m 21 m 11 + m 22 m 12 + m 23 m 13 = 0,
m 21 m 21 + m 22 m 22 + m 23 m 23 = 1, (6.9)

m 21 m 31 + m 22 m 32 + m 23 m 33 = 0,
m 31 m 11 + m 32 m 12 + m 33 m 13 = 0,
m 31 m 21 + m 32 m 22 + m 33 m 23 = 0,
m 31 m 31 + m 32 m 32 + m 33 m 33 = 1. (6.10)

Let the vectors r 1 , r 2 , and r 3 stand for the rows of M :

r 1 = m 11 m 12 m 13 ,

r 2 = m 21 m 22 m 23 ,

r 3 = m 31 m 32 m 33 ,

2
4 −

3

r 1 −

5

M = − r 2 − .

− r 3 −

Now we can rewrite the nine equations more compactly:

r 1 r 1 = 1, r 1 r 2 = 0, r 1 r 3 = 0,

Conditions satisfied by
an orthogonal matrix

r 2 r 1 = 0, r 2 r 2 = 1, r 2 r 3 = 0,

r 3 r 1 = 0, r 3 r 2 = 0, r 3 r 3 = 1.

This notational changes makes it easier for us to make some interpretations.

• First, the dot product of a vector with itself is 1 if and only if the

vector is a unit vector. Therefore, the equations with a 1 on the right-
hand side of the equals sign (Equations (6.8), (6.9), and (6.10)) will
be true only when r 1 , r 2 , and r 3 are unit vectors.

• Second, recall from Section 2.11.2 that the dot product of two vectors

is 0 if and only if they are perpendicular. Therefore, the other six
equations (with 0 on the right-hand side of the equals sign) are true
when r 1 , r 2 , and r 3 are mutually perpendicular.

So, for a matrix to be orthogonal, the following must be true: