3 lesson plans for algebra - Essay Example

The purpose of this lesson is to introduce students to quaternions in order to illustrate a non-commutative operation. The lesson will help students understand that the commutative property of multiplication is not always shared by other operations. However, there are other operations such as matrix multiplication and quaternion multiplication that are non-commutative. In these types of multiplication, the order of the factors affects the product. Quaternions are vectors x = x0 + x1 i + x2 j + x3 k where the coefficients x0, x1, x2 and x3 are real numbers and 1, i, j, and k are basis vectors.

The product of any quaternions are defined by the following equations: Rules of quaternion operations also include: if a and b are scalars, and m and n are one of the quaternions 1, i, j, or k, then the product (am)(bn) = (ab) (mn). The distributive law also holds: if x = x0 + x1 i and y = y0 + y1 i, then their product is This exercise uses space rotations to arrive at quaternion equations. To demonstrate the rotations, a book is used with one end of a belt held firmly between its pages, while the other end of the belt is held firmly under a pile of books on a table.

Three mutually perpendicular axes are used: the k-axis pointing upwards from the book, the i-axis pointing from the book to the right as you face the table, and the j-axis pointing forwards in the direction from the book to the table. Then an i quaternion is represented by rotating the book 180 degrees about the i axis in the positive sense (if your thumb points along the i-axis, the positive direction of rotation is in the direction your fingers curl). Similarly, the j and k quaternions are represented by 180 degree rotations in the positive direction about the j and k axes, respectively.

A twist in the belt represents -1. Now the defining equations (1-4) of