Quaternion Multiplication, Inverse, Conjugate, and Norm Functions with Implementation Examples

In this article, we will explore fundamental operations of quaternions - powerful mathematical tools particularly useful for describing 3D rotations with greater stability and accuracy compared to traditional Euler angles. The implementation covers core quaternion operations through dedicated functions: The quaternion multiplication function follows the Hamilton product rule, where the product of two quaternions q1 = (w1, x1, y1, z1) and q2 = (w2, x2, y2, z2) is computed as: q1 * q2 = (w1*w2 - x1*x2 - y1*y2 - z1*z2, w1*x2 + x1*w2 + y1*z2 - z1*y2, w1*y2 - x1*z2 + y1*w2 + z1*x2, w1*z2 + x1*y2 - y1*x2 + z1*w2) The inverse operation calculates the multiplicative inverse using the formula q^(-1) = conjugate(q) / norm(q)^2, ensuring proper normalization for rotation applications. The conjugate function simply negates the vector components while keeping the scalar part unchanged: conjugate(q) = (w, -x, -y, -z). The norm function computes the magnitude using the Euclidean norm formula: ||q|| = √(w² + x² + y² + z²). We provide a practical programming example demonstrating how to use quaternions for vector rotation coordinate calculations. The rotation implementation involves converting a vector to a pure quaternion, performing quaternion multiplication with the rotation quaternion and its inverse, then extracting the rotated vector components. This approach avoids gimbal lock and provides smooth interpolation capabilities. Through these implementations, you will gain deeper understanding of quaternion mathematical principles and their practical applications in 3D graphics, robotics, and aerospace systems.