https://file.aiquickdraw.com/mathbot/file/1727113035101kosjorkz.jpg

The given vector equation is:

\[
\begin{bmatrix}
y_1 \\
y_2
\end{bmatrix}
= \alpha
\begin{bmatrix}
x_1 \\
x_2
\end{bmatrix}, \quad \alpha \in \mathbb{R}
\]

This equation describes a line in a 2D plane where \( x_1 \) and \( x_2 \) are constants and \( \alpha \) is a scalar parameter that varies over all real numbers \( \mathbb{R} \). The vector \( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \) defines the direction of the line, and as \( \alpha \) changes, the point \( \begin{bmatrix} y_1 \\ y_2 \end{bmatrix} \) moves along the line in this direction.

### Explanation:
- \( \alpha \) is a parameter that "scales" the vector \( \begin{bmatrix} x_1 \\ x_2 \end{bmatrix} \). If \( \alpha = 0 \), the point is at the origin \( (0, 0) \). As \( \alpha \) increases or decreases, the point \( (y_1, y_2) \) moves in a straight line through the origin, with direction determined by \( x_1 \) and \( x_2 \).
- This is the parametric equation of a straight line passing through the origin, with slope \( \frac{x_2}{x_1} \) (if \( x_1 \neq 0 \)).

Would you like further details on how to sketch this line or how different values of \( \alpha \) affect the line's position?

### Relative Questions:
1. What happens when \( \alpha = 0 \) in this equation?
2. How does changing \( \alpha \) affect the direction and length of the vector?
3. What if \( x_1 = 0 \)? How does that affect the line's equation and graph?
4. Can you express this equation in terms of a Cartesian form \( y = mx + b \)?
5. How would you interpret negative values of \( \alpha \) geometrically?

### Tip:
When solving vector equations, always check the magnitude and direction of the constant vector to understand how the line behaves across different values of the parameter.

To sketch the line for the first set of equations in question 1, the vector equation is given as: [y1, y2]^T = α[x1, x2]^T, where x1 and x2 are constants and α is a scalar parameter that varies over all real numbers.

Explore the vector equation for sketching a line in a 2D plane. This problem involves parametric equations and vector scaling, providing a step-by-step breakdown of the line's direction and movement with respect to a scalar parameter α. Suitable for advanced high school and college students.

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