Q2 - Alg 2 - Hw #06 (quaternions)

Front

What is a quaternion?

Back

A quaternion is a mathematical entity that extends complex numbers and is used to represent rotations in three-dimensional space. It is expressed in the form q = a + bi + cj + dk, where a, b, c, and d are real numbers and i, j, k are the fundamental quaternion units.

Front

What is the conjugate of a quaternion q = a + bi + cj + dk?

Back

The conjugate of a quaternion q = a + bi + cj + dk is given by q* = a - bi - cj - dk.

Front

How do you calculate the magnitude of a quaternion q = a + bi + cj + dk?

Back

The magnitude of a quaternion q = a + bi + cj + dk is calculated as |q| = √(a² + b² + c² + d²).

Front

What is the formula for finding the inverse of a quaternion?

Back

The inverse of a quaternion q = a + bi + cj + dk is given by q⁻¹ = (conjugate) / (magnitude)².

Front

What is the significance of the denominator when finding the inverse of a quaternion?

Back

The denominator, which is the square of the magnitude of the quaternion, ensures that the inverse quaternion is properly scaled and represents the correct rotation.

Front

Calculate the magnitude of the quaternion q = 2 + 3i - 8j.

Back

The magnitude |q| = √(2² + 3² + (-8)²) = √(4 + 9 + 64) = √77.

Front

What is the conjugate of the quaternion q = 2 + 3i - 8j?

Back

The conjugate of the quaternion q = 2 + 3i - 8j is q* = 2 - 3i + 8j.

Front

How do you find the denominator for the inverse of a quaternion?

Back

To find the denominator for the inverse of a quaternion, calculate the square of its magnitude.

Front

What is the inverse of the quaternion q = 3 - 4i?

Back

The inverse of the quaternion q = 3 - 4i is q⁻¹ = (3 + 4i) / 25.

Front

What is the formula for the magnitude of a quaternion in terms of its components?

Back

The formula for the magnitude of a quaternion q = a + bi + cj + dk is |q| = √(a² + b² + c² + d²).