If `(x , y)` and `(x ,y)` are the coordinates of the same point referred to two sets of rectangular axes with the same origin and it `u x+v y ,` where `u` and `v` are independent of `xa n dy` , becomes `V X+U Y ,` show that `u^2+v^2=U^2+V^2dot`

Explanation

To solve the problem, we need to show that if \( u x + v y \) becomes \( V X + U Y \) when coordinates are transformed from \( (x, y) \) to \( (X, Y) \) due to a rotation of axes, then \( u^2 + v^2 = U^2 + V^2 \).

### Step-by-Step Solution:

1. **Understanding the Coordinate Transformation**:
When we rotate the axes by an angle \( \theta \), the new coordinates \( (X, Y) \) can be expressed in terms of the old coordinates \( (x, y) \) as follows:
\[
X = x \cos \theta + y \sin \theta
...