Quaternions and spatial rotation around the "z" axis, then flip it 180 degrees around the "x" axis. Take the other book, flip it 180° around "x" axis first, and 90° clockwise around "z" later. The two books do not end up parallel. This shows that, in general, the composition of two different rotations around two distinct spatial axes will not commute. The vector cross product, used to define the axis–angle representation, does confer an orientation ("handedness") to space: in a three-dimensional vector space, the three vectors in the equation will always form a right-handed set (or a left-handed set, depending on how the cross
Triangle it; this is the exterior angle theorem. The sum of the measures of the three exterior angles (one for each vertex) of any triangle is 360 degrees. Two triangles are said to be "similar" if every angle of one triangle has the same measure as the corresponding angle in the other triangle. The corresponding sides of similar triangles have lengths that are in the same proportion, and this property is also sufficient to establish similarity. Some basic theorems about similar triangles are: Two triangles that are congruent have exactly the same size and shape: all pairs of corresponding interior angles