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The image you've uploaded contains a matrix equation, and it seems to be part of a discussion about the conditions under which the rows of a matrix lie in a plane perpendicular to a certain vector. Let's break it down:

### Given Information:
1. You have a matrix \( A \) composed of columns \( a_1, a_2, a_3 \).
2. The equation \( Ax = 0 \) is given, meaning the columns combine to form the zero vector.
3. The system of equations is written as follows:

   \[
   \begin{bmatrix}
   a_1 & a_2 & a_3 \\
   \end{bmatrix}
   \begin{bmatrix}
   x_1 \\
   x_2 \\
   x_3 \\
   \end{bmatrix}
   = 
   \begin{bmatrix}
   0 \\
   0 \\
   0 \\
   \end{bmatrix}
   \]

   This implies that for each row \( r \) of \( A \), we have \( r \cdot x = 0 \).

### Explanation:
- The equation \( r \cdot x = 0 \) suggests that each row vector \( r \) is perpendicular to the vector \( x \). In the context of this problem, \( r_1 \cdot x = 0 \), \( r_2 \cdot x = 0 \), and \( r_3 \cdot x = 0 \) indicate that all three rows lie in a plane perpendicular to the vector \( x \).

### Why is the plane perpendicular to \( x \)?
- The rows of matrix \( A \) are treated as vectors. When we say that \( r \cdot x = 0 \), it implies that the vector \( r \) is orthogonal to \( x \).
- Since each row vector \( r \) satisfies this condition, they all lie in a plane that is perpendicular to the vector \( x \). This is because a vector \( x \) is orthogonal to any vector that lies in the plane if \( x \) is perpendicular to every vector in that plane.

### Summary:
The three rows also lie in a plane because their dot product with \( x \) equals zero, meaning they are perpendicular to \( x \). This creates a situation where the vector \( x \) is normal (perpendicular) to the plane formed by these row vectors.

### Do you want details or have any questions?
#### Relative Questions:
1. How is the concept of a null space related to the plane formed by the rows of a matrix?
2. What are the conditions for a set of vectors to be linearly dependent?
3. How does the rank of a matrix relate to the number of vectors lying in a plane?
4. What does it mean geometrically for a vector to be perpendicular to a plane?
5. Can you explain the relationship between the row space and the null space of a matrix?

#### Tip:
When working with orthogonal vectors, remember that their dot product will always equal zero. This is a key property used in solving many linear algebra problems related to planes and perpendicularity.

If the columns combine into Ax = 0 then each of the rows has r⋅x = 0. The three rows also lie in a plane. Why is that plane perpendicular to x?

Explore why the plane formed by the rows of a matrix is perpendicular to the vector x in the equation Ax=0. This problem covers key concepts in Linear Algebra, including the dot product, orthogonality, and the relationship between the row space and null space. Suitable for undergraduate students studying Linear Algebra.

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