Quaternion Calculator
Our free algebra calculator solves quaternion problems. Get worked examples, visual aids, and downloadable results. See charts, tables, and visual results.
Reviewed for accuracy by Manoj Kumar, Mathematics Educator
Quaternion Calculator
Calculator
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Formula: q = w + xi + yj + zk where i^2 = j^2 = k^2 = ijk = -1
Worked example — q1 * q2 = -9 + 18i + 3j + 6k | |q1| = 5.4772, |q2| = 3.8730
Formula
q = w + xi + yj + zk where i^2 = j^2 = k^2 = ijk = -1
Quaternion multiplication follows Hamilton rules. For unit quaternions, q = cos(theta/2) + sin(theta/2)(xi + yj + zk) represents rotation by angle theta around axis (x,y,z).
Worked Examples
Example 1: Quaternion Multiplication
Problem:Multiply q1 = 1 + 2i + 3j + 4k by q2 = 2 + i - j + 3k.
Solution:Using the quaternion product formula: w = (1)(2) - (2)(1) - (3)(-1) - (4)(3) = 2 - 2 + 3 - 12 = -9 x = (1)(1) + (2)(2) + (3)(3) - (4)(-1) = 1 + 4 + 9 + 4 = 18 y = (1)(-1) - (2)(3) + (3)(2) + (4)(1) = -1 - 6 + 6 + 4 = 3 z = (1)(3) + (2)(-1) - (3)(1) + (4)(2) = 3 - 2 - 3 + 8 = 6 Result: -9 + 18i + 3j + 6k
Result:q1 * q2 = -9 + 18i + 3j + 6k | |q1| = 5.4772, |q2| = 3.8730
Example 2: Rotation Quaternion to Euler Angles
Problem:Convert the unit quaternion q = 0.7071 + 0.7071i + 0j + 0k to Euler angles.
Solution:This quaternion represents a 90-degree rotation about the x-axis. q = cos(90/2) + sin(90/2)(1i + 0j + 0k) = cos(45) + sin(45)i Euler angles (ZYX convention): Roll = atan2(2(0.7071*0.7071 + 0*0), 1 - 2(0.7071^2 + 0^2)) = atan2(1, 0) = 90 degrees Pitch = asin(2(0.7071*0 - 0*0.7071)) = 0 degrees Yaw = atan2(2(0.7071*0 + 0.7071*0), 1 - 2(0 + 0)) = 0 degrees
Result:Roll = 90 degrees, Pitch = 0 degrees, Yaw = 0 degrees
Frequently Asked Questions
What is a quaternion and why was it invented?
A quaternion is a four-dimensional number of the form q = w + xi + yj + zk, where w, x, y, z are real numbers and i, j, k are imaginary units satisfying i^2 = j^2 = k^2 = ijk = -1. Quaternions were invented by Sir William Rowan Hamilton in 1843 while walking along the Royal Canal in Dublin, Ireland. He was searching for a way to extend complex numbers to three dimensions but discovered that four dimensions were needed. Quaternions provide a compact and computationally efficient way to represent 3D rotations without the singularity problems (gimbal lock) that plague Euler angle representations.
How does quaternion multiplication work?
Quaternion multiplication follows specific rules derived from the fundamental relations i^2 = j^2 = k^2 = ijk = -1, which also imply ij = k, jk = i, ki = j and their reverses ji = -k, kj = -i, ik = -j. For two quaternions q1 = (w1, x1, y1, z1) and q2 = (w2, x2, y2, z2), the product has: w = w1*w2 - x1*x2 - y1*y2 - z1*z2, x = w1*x2 + x1*w2 + y1*z2 - z1*y2, y = w1*y2 - x1*z2 + y1*w2 + z1*x2, z = w1*z2 + x1*y2 - y1*x2 + z1*w2. Crucially, quaternion multiplication is not commutative, meaning q1*q2 does not generally equal q2*q1.
How do quaternions represent 3D rotations?
A unit quaternion (magnitude = 1) of the form q = cos(theta/2) + sin(theta/2)(xi + yj + zk) represents a rotation of angle theta around the axis defined by the unit vector (x, y, z). To rotate a point p = (px, py, pz), you express it as a pure quaternion P = 0 + px*i + py*j + pz*k, then compute the rotated point as P_rotated = q * P * q_conjugate. The resulting pure quaternion gives the rotated coordinates. This representation has several advantages: it uses only 4 numbers instead of 9 (rotation matrix), it is easy to normalize, and composing rotations is simply quaternion multiplication.
What is gimbal lock and how do quaternions avoid it?
Gimbal lock is a phenomenon that occurs with Euler angles when two rotation axes align, causing a loss of one degree of freedom. For example, if pitch reaches 90 degrees in the ZYX convention, the roll and yaw axes become parallel, and the system can only rotate around two independent axes instead of three. This creates sudden jumps and singularities in animation and control systems. Quaternions avoid gimbal lock entirely because they represent rotations as a single operation in four-dimensional space rather than three sequential rotations around fixed axes. This is why quaternions are the preferred rotation representation in game engines, aerospace, and robotics.
What is the conjugate and inverse of a quaternion?
The conjugate of a quaternion q = w + xi + yj + zk is q* = w - xi - yj - zk, obtained by negating the vector (imaginary) part while keeping the scalar (real) part unchanged. The magnitude of q is |q| = sqrt(w^2 + x^2 + y^2 + z^2). The inverse of q is q^(-1) = q* / |q|^2, which is the conjugate divided by the squared magnitude. For unit quaternions (|q| = 1), the inverse equals the conjugate, which makes computations particularly efficient. The conjugate of a rotation quaternion represents the reverse rotation, and q * q* = |q|^2 always equals a real number.
How do you convert between quaternions and Euler angles?
Converting from Euler angles (roll, pitch, yaw) to a quaternion involves computing half-angles and using trigonometric products. For the ZYX convention: w = cos(r/2)*cos(p/2)*cos(y/2) + sin(r/2)*sin(p/2)*sin(y/2), and similar expressions for x, y, z components. Converting back from quaternion to Euler angles uses arctangent and arcsine functions on combinations of quaternion components. The reverse conversion has a singularity when pitch equals plus or minus 90 degrees (corresponding to gimbal lock in Euler angles). Different software packages may use different Euler angle conventions, so it is essential to verify which convention is expected.
What is spherical linear interpolation (SLERP) for quaternions?
SLERP is a method for smoothly interpolating between two quaternion rotations, producing a constant-speed rotation along the shortest arc on the four-dimensional unit sphere. Given quaternions q1 and q2 and parameter t from 0 to 1, SLERP computes: slerp(q1, q2, t) = q1 * sin((1-t)*omega) / sin(omega) + q2 * sin(t*omega) / sin(omega), where omega is the angle between q1 and q2 computed as cos(omega) = q1 dot q2. SLERP is essential in computer animation for creating smooth camera movements and character joint rotations. Unlike linear interpolation of Euler angles, SLERP guarantees constant angular velocity and no unwanted wobble.
How do quaternions relate to rotation matrices?
Every unit quaternion corresponds to a unique 3x3 rotation matrix, and every rotation matrix can be converted to a quaternion (with a sign ambiguity since q and -q represent the same rotation). The rotation matrix R from quaternion q = (w, x, y, z) has entries involving products of quaternion components. For example, R[0][0] = 1 - 2(y^2 + z^2) and R[0][1] = 2(xy - wz). Quaternions are more compact (4 vs 9 numbers), easier to normalize (divide by magnitude vs Gram-Schmidt orthogonalization), and composing rotations is cheaper (16 multiplications for quaternion product vs 27 for matrix multiplication). However, applying a rotation to a vector is slightly faster with matrices.
What are dual quaternions and what are they used for?
Dual quaternions extend regular quaternions by combining rotation and translation into a single algebraic entity. A dual quaternion has the form q = q_real + epsilon * q_dual, where epsilon is the dual number satisfying epsilon^2 = 0. The real part encodes rotation and the dual part encodes translation. Dual quaternions provide the most compact representation of rigid body transformations (8 numbers vs 12 for a transformation matrix) and support smooth interpolation of both rotation and translation simultaneously via ScLERP. They are widely used in character animation for skinning (blending multiple bone transformations), robotics for kinematics chains, and computer vision for camera pose estimation.
What is the history and mathematical significance of quaternions?
Quaternions have a rich mathematical history dating to 1843 when Hamilton carved the fundamental formula i^2 = j^2 = k^2 = ijk = -1 into Brougham Bridge in Dublin. They were the first example of a non-commutative algebra, challenging the assumption that multiplication must be commutative. Quaternions dominated vector analysis throughout the late 19th century before Gibbs and Heaviside promoted the simpler vector notation we use today. However, quaternions experienced a renaissance in the late 20th century with the rise of computer graphics, robotics, and aerospace engineering. They form a normed division algebra, and by the Frobenius theorem, they are one of only four such algebras (along with reals, complex numbers, and octonions).
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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