Question

Let \(\alpha=3+4 i+7 j+k\) and \(\beta=2-3 i+j-k\) be quaternions. Calculate \(\alpha \beta\) and \(\alpha / \beta\).

Short answer

Question: Calculate the product and division of the quaternions \(\alpha = 3 + 4i + 7j + k\) and \(\beta = 2 - 3i + j - k\). Answer: The product of the quaternions is \(\alpha \beta = -23 - 7i - 9j + 9k\), and the division of the quaternions is \(\alpha / \beta = -1 + \frac{17}{15}i + \frac{2}{5}k\).

Step-by-step solution

Review quaternion multiplication rules

Quaternions are an extension of complex numbers and consist of a scalar part and a vector part. The quaternion multiplication follows these rules: 1. Scalars multiply with scalars and vectors using the distributive property. 2. The product of two vector parts follows the rules of cross product. 3. \(i^2 = j^2 = k^2 = ijk = -1\)

Multiply \(\alpha\) and \(\beta\)

Using the rules from Step 1, we can multiply the quaternions: \(\alpha \beta = (3+4i+7j+k)(2-3i+j-k)\) Now, expand the product and combine the terms: \(\alpha \beta = 6 - 9i + 3j - k + 8i^2 - 12ij + 4ik + 14ij - 21i^2 + 7j^2 - k^2 + 2ik - 3k^2\)

Simplify the product

We need to apply the rules for simplifying quaternion products: \(\alpha \beta = 6 - 9i + 3j - k - 8 - 12ij + 4ik - 21 - 7 - k^2 + 2ik + 3k^2\) Combine the like terms: \(\alpha \beta = -23 + (-9+2)i+ (-12+3)j+(4-1+6)k\) Therefore, \(\alpha \beta = -23 - 7i - 9j + 9k\)

Find the inverse of \(\beta\)

The inverse of quaternion \(\beta\) can be found by dividing the conjugate of \(\beta\) by the squared norm of \(\beta\): \(\beta^{-1} = \frac{\text{Conj}(\beta)}{||\beta||^2}\) The conjugate of \(\beta\) is: \(\text{Conj}(\beta) = 2+3i-j+k\) The squared norm of \(\beta\) is: \(||\beta||^2 = 2^2+(-3)^2+1^2+(-1)^2=4+9+1+1=15\) So, the inverse of \(\beta\) is: \(\beta^{-1}=\frac{1}{15}(2+3i-j+k)\)

Calculate \(\alpha / \beta\)

We can now find the division \(\alpha / \beta\) by multiplying \(\alpha\) with \(\beta^{-1}\): \(\alpha / \beta = \alpha \cdot \beta^{-1} = (3+4i+7j+k) \cdot \frac{1}{15}(2+3i-j+k)\) Now, expand the product, apply the rules from Step 1 and simplify: \(\alpha / \beta = \frac{1}{15}(6+9i-3j+3k+8i^2-12ij+4ik+14ij-7j^2-21i^2+k^2+2ik-3k^2) \) \(\alpha / \beta= \frac{1}{15}(-15+17i-k+7k)\) Finally, \(\alpha / \beta = -1 + \frac{17}{15}i + \frac{6}{15}k\) Which can be written as: \(\alpha / \beta = -1 + \frac{17}{15}i + \frac{2}{5}k\)

Key concepts

Quaternion Multiplication

Quaternions extend complex numbers by introducing three imaginary units: \( i, j, \) and \( k \). Like complex numbers, quaternions have a real part and imaginary parts, but they expand into three dimensions. This makes quaternion multiplication unique. Here's how it works:

• Use the distributive property. Multiply each part of the first quaternion by each part of the second.
• Combine like terms, considering that \( i^2 = j^2 = k^2 = ijk = -1 \).
• Cross product rules apply to imaginary units when multiplying (e.g., \( ij = k \)).

When multiplying quaternions \( \alpha = 3 + 4i + 7j + k \) and \( \beta = 2 - 3i + j - k \), start by expanding: \[ \alpha \beta = (3+4i+7j+k)(2-3i+j-k) \] Expand and simplify using the rules to find the resulting quaternion product.

Quaternion Division

Division of quaternions is not as straightforward as division of real numbers. It involves multiplication with an inverse. Here's how to proceed:

• First, find the inverse of the divisor quaternion.
• Multiply the dividend quaternion by this inverse.

To calculate \( \alpha / \beta \), where \( \alpha = 3 + 4i + 7j + k \) and \( \beta = 2 - 3i + j - k \), first find \( \beta^{-1} \), and then compute \( \alpha \cdot \beta^{-1} \). Simplifying the expression will yield the division result. This approach ensures the division respects quaternion properties.

Quaternion Inverse

To divide quaternions, you need the inverse of the divisor. The inverse of a quaternion is like a reciprocal in the sense of division. Here is the process:

• Find the conjugate of the quaternion. The conjugate of \( \beta = 2 - 3i + j - k \) is \( \text{Conj}(\beta) = 2 + 3i - j + k \).
• Calculate the norm squared: \( ||\beta||^2 = 2^2 + (-3)^2 + 1^2 + (-1)^2 = 15 \).
• Now, use the formula for the inverse: \( \beta^{-1} = \frac{\text{Conj}(\beta)}{||\beta||^2} \).

With this inverse, you can perform division by multiplying the original quaternion by the inverse.

Complex Numbers Extension

Quaternions extend complex numbers by introducing three dimensions of imaginary numbers. This extension enriches the mathematical toolkit, allowing for calculations in four-dimensional space. While complex numbers use just one imaginary unit \( i \), quaternions use \( i, j, \) and \( k \).

• Quaternion rules like \( i^2 = j^2 = k^2 = ijk = -1 \) provide new ways to approach problems.
• They are extremely useful in three-dimensional computations like rotations in computer graphics.

By understanding the foundation of quaternions, you can tackle more complex scenarios than those solvable with just complex numbers. This makes them a powerful tool in advanced mathematics and physics.