Problem 15 - Classical Dynamics of Particles and Systems

Question

Find the values of \(\alpha\) needed to make the following transformation orthogonal. $$ \left(\begin{array}{rrr} 1 & 0 & 0 \\ 0 & \alpha & -\alpha \\ 0 & \alpha & \alpha \end{array}\right) $$

Step-by-Step Solution

Answer

The values of α needed to make the given transformation orthogonal are \( \alpha = \pm \frac{1}{\sqrt{2}} \).

Step 1: Find the dot product of all row pairs and norms of the rows

In the given matrix, the three rows are: 1. Row 1: \((1,0,0)\) 2. Row 2: \((0,\alpha,-\alpha)\) 3. Row 3: \((0,\alpha,\alpha)\) We need to evaluate the dot products between all pairs of rows as well as the norms of each of the rows. Dot products: 1. Row 1 \(\cdot\) Row 2: \(1*0 + 0*\alpha + 0*(-\alpha) = 0\) 2. Row 1 \(\cdot\) Row 3: \(1*0 + 0*\alpha + 0*\alpha = 0\) 3. Row 2 \(\cdot\) Row 3: \(0*0 + \alpha*\alpha - \alpha*\alpha = 0\) Norms: 1. \( \lVert \text{Row }1 \rVert = \sqrt{1^2 + 0^2 + 0^2} = 1 \) 2. \( \lVert \text{Row }2 \rVert = \sqrt{0^2 + \alpha^2 + (-\alpha)^2} = \sqrt{2\alpha^2} \) 3. \( \lVert \text{Row }3 \rVert = \sqrt{0^2 + \alpha^2 + \alpha^2} = \sqrt{2\alpha^2} \)

Step 2: Evaluate the conditions for orthogonality

Now we have all the dot products and norms to check the conditions for orthogonal transformation: 1. Row 1 \(\cdot\) Row 2 = 0 (condition already met) 2. Row 1 \(\cdot\) Row 3 = 0 (condition already met) 3. Row 2 \(\cdot\) Row 3 = 0 (condition already met) 4. Norm Row 1 = 1 (condition already met) 5. Norm Row 2 = 1, so we need: \( \sqrt{2\alpha^2}=1 \) 6. Norm Row 3 = 1, so we need: \( \sqrt{2\alpha^2}=1 \) (same condition as row 2) From the 5th and 6th conditions, we only need to solve one equation for α.

Step 3: Solve the equation for α

To find the values of α, we solve the equation: \( \sqrt{2\alpha^2}=1 \) Square both sides: \( 2\alpha^2 =1 \) Now, divide by 2: \( \alpha^2 = \frac{1}{2} \) Finally, take the square root: \( \alpha = \pm \frac{1}{\sqrt{2}} \) Thus, the values of α needed to make the given transformation orthogonal are \( \alpha = \pm \frac{1}{\sqrt{2}} \).

Key Concepts

Dot Product

The dot product is a way of multiplying two vectors that gives a scalar result. Simply put, it's an arithmetic operation you can use to determine the similarity between two vectors based on their direction. This is extremely useful in many areas of math and physics, where understanding how similar vectors are can tell us a lot about the system we are dealing with.
To calculate the dot product, you multiply each corresponding component of the vectors and then sum the results. For example, for two vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), the dot product is given by:
\(\mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 + a_3b_3 \) This operation helps in determining orthogonality: when the dot product of two vectors is zero, they are orthogonal, meaning the vectors are at a 90-degree angle to each other. This aspect of vectors helps to simplify and solve many problems relating to orthogonal transformations.

Matrix Norm

In linear algebra, the norm of a vector or matrix is a measure of its "size" or "length." It captures the overall magnitude of a matrix or a vector and is a crucial concept when dealing with orthogonality and many other matrix operations.
The norm most commonly used in vectors and matrices is called the Euclidean norm, often represented by \( \lVert \cdot \rVert \). For a vector \( \mathbf{v} = (v_1, v_2, ..., v_n) \), its norm is calculated as:
\[\lVert \mathbf{v} \rVert = \sqrt{v_1^2 + v_2^2 + ... + v_n^2}\]
Similarly, when calculating the norm of each row vector of the matrix, it helps determine if the vectors are unit vectors, which is a requirement for orthogonality in transformations. We can use the norm to check if each row is of unit length, i.e., if \( \lVert \text{Row } i \rVert = 1\), thereby maintaining orthogonality. If not, adjustments like solving for alpha (as shown in the solution) are necessary to make the transformation orthogonal.

Square Root

The square root is a fundamental arithmetic operation linked to solving many mathematical problems, especially those involving quadratic equations, such as in our exercise where we derive the values of \( \alpha \) for orthogonality. Taking the square root involves finding a number that, when multiplied by itself, gives the original number. In math terms, if \( x = \sqrt{y} \), then \( x \cdot x = y \).
This operation is crucial in finding the norms of vectors, as shown when determining whether the norm \( \sqrt{2\alpha^2} = 1 \) leads to the solution of what \( \alpha \) must be. In our problem, calculating the square root of both components and working through transformation conditions helps in arriving at the necessary values for an orthogonal matrix, simplifying calculations and solutions.