Let's define SE(3) = SO(3) * Ras in the class. In other words, we define an ordered pair of elements (R,x) E SE(3), of which the binary operator is given as (R1, X1) (R2, X2) = (R_R2, Rx2 + x1). As we learned, this is called a "semi-direct product". (a) Prove that SE(3) is a group - but refrain from using the homogeneous representation to prove this. (b) Prove that SE(3) has a well defined action on points in R3 given by g(p) = Rp + x, where g=(R,x) E SE(3) and p ER3. (c) Prove that the induced action on vectors, 9+(v) = g(v + p) - 9(p) is given by g+(v) = Rv. Let's define SE(3) = SO(3) * Ras in the class. In other words, we define an ordered pair of elements (R,x) E SE(3), of which the binary operator is given as (R1, X1) (R2, X2) = (R_R2, Rx2 + x1). As we learned, this is called a "semi-direct product". (a) Prove that SE(3) is a group - but refrain from using the homogeneous representation to prove this. (b) Prove that SE(3) has a well defined action on points in R3 given by g(p) = Rp + x, where g=(R,x) E SE(3) and p ER3. (c) Prove that the induced action on vectors, 9+(v) = g(v + p) - 9(p) is given by g+(v) = Rv