Question

Show that the determinant is equal to 1. Note that this is the rotational matrix for a unit quaternion q = a + bi + cj + dk.

bc+ac

Answer 1

The determinant matrix is equal to 1 and Dat of Rotation matrix is equal to 1 to show that r is rotation matrix we have to show that Determine of R is 1 given that R = [2 (a^2 + b^2) - 1 2 (bc - ad) 2 (db + ac) 2 (bc + ad) 2 (a^2 + c^2) - 1 2(cd - ab) 2(db - ac) 2 (cd + ab) 2 (a^2 + b^2) - 1] Vertical-bar R Vertical-bar = [2 (a^+ b^2)] [(2a^2 + 2c^2 - 1) (2a^2 + 2d^2) - (2 cd + 2ab) (2 cd - 2ba)] = + 2 (db + ac) [2 (bc + 2ad) (2 cd + 2 ab) - (2 bd - 2 ac) (2 a^2 + 2c^2 - 1] = [2 a^2 + 2 - 1 4a^2 + 4 a^b^2 - 2a^2 + 4a^c^+ c^d^- 2 c^2 - 2a^2 - 2d^2 + 1 - 4 c^2d^2 + 4abcd - 4 abcd + 4 a^2 b^2]