Math Problem Statement

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.

Solution

The given vector equation is:

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.

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Math Problem Analysis

Mathematical Concepts

Vector Equation
Parametric Equations
Linear Algebra

Formulas

y1 = α * x1
y2 = α * x2
Vector equation [y1, y2]^T = α * [x1, x2]^T

Theorems

Parametric form of a line in 2D
Vector scaling and direction

Suitable Grade Level

Grades 11-12, College Level