Solution Analysis

We are given the following conditions:

• Condition 1: $x^2 + y^2 = 1$
• Condition 2: $u^2 + v^2 = 1$
• Condition 3: $xu + yv = 0$

We need to determine which of the following statements are true:

1. $x^2 + u^2 = 1$
2. $y^2 + v^2 = 1$
3. $xy + uv = 0$

Verifying Statement 1: $x^2 + u^2 = 1$

From Condition 3, $xu = -yv$. Squaring both sides gives $x^2u^2 = y^2v^2$.

From Condition 1, $y^2 = 1 - x^2$.

From Condition 2, $v^2 = 1 - u^2$.

Substitute these into the squared equation:

$x^2u^2 = (1 - x^2)(1 - u^2)$

$x^2u^2 = 1 - u^2 - x^2 + x^2u^2$

Subtract $x^2u^2$ from both sides:

$0 = 1 - u^2 - x^2$

Rearranging gives $x^2 + u^2 = 1$.

Therefore, statement 1 is true.

Verifying Statement 2: $y^2 + v^2 = 1$

Using the result from Statement 1, $x^2 + u^2 = 1$.

From Condition 1, $x^2 = 1 - y^2$.

From Condition 2, $u^2 = 1 - v^2$.

Substitute these into the equation $x^2 + u^2 = 1$:

$(1 - y^2) + (1 - v^2) = 1$

$2 - y^2 - v^2 = 1$

Rearranging gives $y^2 + v^2 = 2 - 1$.

$y^2 + v^2 = 1$.

Therefore, statement 2 is true.

Verifying Statement 3: $xy + uv = 0$

Condition 3, $xu + yv = 0$, indicates that the vectors $(x, y)$ and $(u, v)$ are orthogonal.

Since $(x, y)$ and $(u, v)$ are unit vectors (from Conditions 1 and 2), the vector $(u, v)$ must be a rotation of $(x, y)$ by +/- 90 degrees, scaled by -1 if needed.

This means either $(u, v) = (-y, x)$ or $(u, v) = (y, -x)$.

Case 1: $u = -y$ and $v = x$.

Substitute into Statement 3: $xy + uv = xy + (-y)(x) = xy - xy = 0$.

Case 2: $u = y$ and $v = -x$.

Substitute into Statement 3: $xy + uv = xy + (y)(-x) = xy - xy = 0$.

In both cases, Statement 3 is true.

Conclusion

All three statements (1, 2, and 3) are true based on the given conditions.