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The problem asks to verify that the vectors \(\vec{v}_1 = (-y, x)\) and \(\vec{v}_2 = (y, -x)\) are perpendicular to the vector \(\vec{v} = (x, y)\).

To check if two vectors are perpendicular, we calculate their dot product. If the dot product is zero, the vectors are perpendicular.

### 1. Check if \(\vec{v}_1\) is perpendicular to \(\vec{v}\):
\[
\vec{v}_1 \cdot \vec{v} = (-y) \cdot x + x \cdot y = -yx + xy = 0
\]
Thus, \(\vec{v}_1\) is perpendicular to \(\vec{v}\).

### 2. Check if \(\vec{v}_2\) is perpendicular to \(\vec{v}\):
\[
\vec{v}_2 \cdot \vec{v} = y \cdot x + (-x) \cdot y = yx - xy = 0
\]
Thus, \(\vec{v}_2\) is perpendicular to \(\vec{v}\).

Since both dot products are zero, both vectors \(\vec{v}_1\) and \(\vec{v}_2\) are perpendicular to \(\vec{v}\).

Would you like further details or have any other questions?

Here are 5 related questions for further exploration:
1. What is the geometric interpretation of the dot product of two vectors?
2. How do you calculate the angle between two vectors using the dot product?
3. What are the conditions for vectors to be orthogonal (perpendicular)?
4. How do you find a unit vector perpendicular to a given vector in 2D?
5. How does the cross product differ from the dot product in terms of vector relationships?

**Tip:** For higher-dimensional vectors, the dot product can be extended in the same way to check for perpendicularity.

Comprueba que los vectores \(\vec{v}_1 = (-y, x)\) y \(\vec{v}_2 = (y, -x)\) son perpendiculares al vector \(\vec{v} = (x, y)\).

Learn how to verify that the vectors \(\vec{v}_1 = (-y, x)\) and \(\vec{v}_2 = (y, -x)\) are perpendicular to \(\vec{v} = (x, y)\) by using the dot product. A detailed solution is provided using vector algebra concepts. Suitable for high school students in Grades 10-12.

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