**3.c) Show that:i) Two successive rotations are additiveii) Two successive scalings are multiplicative**

**Answer:**

**i) Two successive rotations are additive**

Two successive rotations applied to a point P produce the transformed position

P = R(θ2) · {R(θ1) · P}

= {R(θ2) · R(θ1)} · P

By multiplying the two rotation matrices, we can verify that two successive rotations are additive:

R(θ2) · R(θ1) = R(θ1 + θ2)

so that the final rotated coordinates of a point can be calculated with the composite rotation matrix as

P’ = R(θ1 + θ2) · P

**ii) Two successive scalings are multiplicative**

Concatenating transformation matrices for two successive scaling operations in two dimensions produces the following composite scaling matrix:

The resulting matrix in this case indicates that successive scaling operations are multiplicative. That is, if we were to triple the size of an object twice in succession, the final size would be nine times that of the original.