Find a parametrization of a hyperplane in $\mathbb{R}^4$ given by the equation $x+y+z+at=b$

Question

Find a parametrization of the hyperplane in $\mathbb{R}^4$ given by the equation $x+y+z+at=b$ where $a,b$ are real numbers.

I'm not sure about my answer:

$$y \begin{pmatrix} -1\\ 1\\ 0\\ 0 \end{pmatrix} + x \begin{pmatrix} -1\\ 0\\ 1\\ 0 \end{pmatrix} +t \begin{pmatrix} -a\\ 0\\ 0\\ 1 \end{pmatrix} + b \begin{pmatrix} 1\\ 0\\ 0\\ 0 \end{pmatrix}$$

Is this correct?

Thanks!

Answer 1

Your parametrization is correct.

Just to make my answer a little more substantial, I'll tell you that I'd tweak your answer a bit. $(1)$ I'd name the parameterization. $(2)$ I'd introduce new variable names for the parameters. And $(3)$ I'd specify the values that those new variables can take. So I'd write the answer as

$$\vec v(p,q,r) = \pmatrix{x \\ y \\ z \\ t} = p \begin{pmatrix} -1\\ 1\\ 0\\ 0 \end{pmatrix} + q \begin{pmatrix} -1\\ 0\\ 1\\ 0 \end{pmatrix} +r \begin{pmatrix} -a\\ 0\\ 0\\ 1 \end{pmatrix} + \begin{pmatrix} b\\ 0\\ 0\\ 0 \end{pmatrix},\quad p,q,r\in \Bbb R$$

If you can explain why you're not confident about your solution in the comments below, I'll attempt to assuage your doubts.