# Complete Solution in Structured Format --- ## 1. Complete Solution with All Calculations ### (a) Line \( l \subseteq \mathbb{R}^2 \) with equation \( 2x + y = 3 \): **Step 1:** Express \( y \) in terms of \( x \): \[ 2x + y = 3 \implies y = 3 - 2x \] **Step 2:** Express the general point on the line as a vector: \[ \vec{r} = \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} x \\ 3-2x \end{pmatrix} \] **Step 3:** Write in vector form using a parameter \( t \in \mathbb{R} \): \[ \vec{r} = \begin{pmatrix} t \\ 3-2t \end{pmatrix} \] Or equivalently, \[ \vec{r} = \begin{pmatrix} 0 \\ 3 \end{pmatrix} + t \begin{pmatrix} 1 \\ -2 \end{pmatrix} \] --- ### (b) Plane \( P \subseteq \mathbb{R}^3 \) with equation \( x + y + z = 1 \): **Step 1:** Express \( z \) in terms of \( x \) and \( y \): \[ x + y + z = 1 \implies z = 1 - x - y \] **Step 2:** Express a general point on the plane as a vector: \[ \vec{r} = \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} x \\ y \\ 1-x-y \end{pmatrix} \] **Step 3:** Write in vector form using parameters \( s, t \in \mathbb{R} \): \[ \vec{r} = \begin{pmatrix} s \\ t \\ 1-s-t \end{pmatrix} \] Or equivalently, \[ \vec{r} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} + s \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix} + t \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix} \] --- ## 2. Explanation of Each and Every Part ### Part (a): Line in \( \mathbb{R}^2 \) - The equation \( 2x + y = 3 \) represents a straight line in the plane. - Solving for \( y \) gives the relationship between \( x \) and \( y \) for any point on the line. - By letting \( x = t \) (where \( t \) is a free real parameter), the vector form is established. - The vector form shows the line passes through the point \( (0,3) \) and has direction vector \( (1,-2) \). ### Part (b): Plane in \( \mathbb{R}^3 \) - The equation \( x + y + z = 1 \) represents a plane in three-dimensional space. - Solving for \( z \) gives the relationship among \( x \), \( y \), and \( z \). - By letting \( x = s \) and \( y = t \) (where \( s \) and \( t \) are free real parameters), the vector form for the plane is derived. - The vector form shows the plane passes through \( (0,0,1) \) and can be spanned by the vectors \( (1,0,-1) \) and \( (0,1,-1) \). --- ## 3. Final Conclusion Statement - The line \( l \subseteq \mathbb{R}^2 \) with equation \( 2x + y = 3 \) can be expressed in vector form as \( \vec{r} = \begin{pmatrix} 0 \\ 3 \end{pmatrix} + t \begin{pmatrix} 1 \\ -2 \end{pmatrix} \), where \( t \in \mathbb{R} \). - The plane \( P \subseteq \mathbb{R}^3 \) with equation \( x + y + z = 1 \) can be expressed in vector form as \( \vec{r} = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} + s \begin{pmatrix} 1 \\ 0 \\ -1 \end{pmatrix} + t \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix} \), where \( s, t \in \mathbb{R} \). - Both forms provide a parametric representation of the given line and plane, respectively, in vector notation.