Problem 241 - Mathematics for IIT JEE Main and Advanced Differential Calculus Algebra Trigonometry

Question

If \(F(\alpha)=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0 \\ \sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1\end{array}\right]\) then show that \(F(x) F(y)=F(x+y)\). Hence prove that \([F(x)]^{-1}=F(-x)\).

Step-by-Step Solution

Answer

Upon performing the matrix calculations and applying the formulas for sine and cosine, it can be seen that F(x)F(y) yields the same matrix as F(x+y), proving the first part of the problem. Similarly, the matrix obtained for [F(x)]^-1 matches with F(-x), proving the second part of the problem.

Step 1: Compute F(x)F(y)

To compute F(x)F(y), perform a matrix multiplication. Given F(x) = [[cosx, -sinx, 0], [sinx, cosx, 0], [0, 0, 1]] and F(y) = [[cosy, -siny, 0], [siny, cosy, 0], [0, 0, 1]], their multiplication will result in a new 3x3 matrix.

Step 2: Compute F(x+y)

To compute F(x+y), you simply replace \(\alpha\) in F(\(\alpha\)) with \(x+y\). This will result in F(x+y) = [[cos(x+y), -sin(x+y), 0], [sin(x+y), cos(x+y), 0], [0, 0, 1]]. Use the formulas for sine and cosine of a sum: sin(x + y) = sinx cosy + cosx siny and cos(x + y) = cosx cosy - sinx siny.

Step 3: Compare F(x)F(y) and F(x+y)

Compare the matrix obtained from F(x)F(y) multiplication with F(x+y). If they are equivalent, then F(x)F(y) = F(x+y).

Step 4: Compute [F(x)]^-1

To prove that [F(x)]^-1 = F(-x), first compute the inverse of F(x). [F(x)]^-1 of the matrix can be found using the method for finding the inverse of a matrix. For this particular matrix, the inverse matrix exists because the determinant is not zero.

Step 5: Compute F(-x)

Then calculate F(-x) by replacing \(\alpha\) in F(\(\alpha\)) with -x. The output will be a new 3x3 matrix.

Step 6: Compare [F(x)]^-1 and F(-x)

Compare the matrix obtained for [F(x)]^-1 with F(-x). If they are equivalent, then [F(x)]^-1 = F(-x), thereby proving the given condition.

Key Concepts

Rotation Matrix

A rotation matrix is a powerful tool in mathematics and engineering. It allows us to perform rotation transformations in the Euclidean space. In the context of two-dimensional or three-dimensional spaces, this becomes especially useful. A 3D rotation matrix, like the one given in the problem, typically involves an angle of rotation \(\alpha\), and works around a specific axis.
This matrix can be used to rotate points around various axes while maintaining their relative distances. For instance, the provided matrix rotates points in 3D around the z-axis by an angle \(\alpha\). Its structure usually consists of cosine and sine functions that ensure the distances and angles between points remain constant after transformation.

• The elements employ trigonometric identities like \(\cos \alpha\) and \(-\sin \alpha\)
• The axes not involved in the rotation (z-axis, in this case) maintain a standard identity operation
• This particular form is crucial in computer graphics, robotics, and physics

Inverse Matrix

An inverse matrix, when multiplied with its original matrix, results in an identity matrix. This concept is fundamental in solving linear equations and transformations. In the exercise, we're tasked with proving that the inverse of a rotation matrix \(F(x)\) is equal to \(F(-x)\).
The inverse of a matrix is only possible when the determinant of the matrix is non-zero. For rotation matrices, this condition is typically met because they are orthogonal and their determinant is one.

• To find an inverse, rearrange components and apply a negative sign to the angles
• The operation yields a matrix that counteracts the effect of the original transformation
• Rotating by an angle \(x\) followed by \(-x\) brings you back to the original position

Trigonometric Identities

Trigonometric identities serve as essential tools in simplifying expressions involving trigonometric functions. In the context of the exercise, we use these identities to simplify the product of two rotation matrices, \(F(x)F(y)\), and to show they equal \(F(x+y)\), exploiting identities such as:

• \(\sin(x+y) = \sin x \cos y + \cos x \sin y\)
• \(\cos(x+y) = \cos x \cos y - \sin x \sin y\)

These properties allow us to decompose complex trigonometric functions into a format that mirrors the structure of a new rotation matrix. It demonstrates how the cumulative effect of two rotations is equivalent to a single rotation by the sum of the angles.

Matrix Algebra

Matrix algebra encompasses operations such as addition, subtraction, scalar multiplication, and the critical operation of matrix multiplication. Here, we delve into this last operation to find \(F(x)F(y)\). The key is understanding that matrix multiplication is not merely an element-wise operation but a calculation involving the rows and columns of the matrices involved.

• Each entry in the resulting matrix comes from dot products of rows and columns
• This multiplication respects the non-commutative nature of matrices: \(AB eq BA\)
• Using identity elements like the zeroes in non-rotational axes simplifies the computation

In our exercise, this ability to multiply matrices and compare them with another matrix, \(F(x+y)\), is crucial in proving equivalency in transformations.