# Find a set of vectors that spans the plane P1: 2x + 3y -z = 0

Question
Vectors and spaces
Find a set of vectors that spans the plane $$\displaystyle{P}{1}:{2}{x}+{3}{y}-{z}={0}$$

2021-01-06
Note that if $$\displaystyle{\left({x},{y},{z}\right)}∈{P}{1}$$ then $$\displaystyle{2}{x}+{3}{y}−{z}={0}.$$ Therefore we have $$\displaystyle{z}={2}{x}+{3}{y}$$
$$\displaystyle{\left({x},{y},{z}\right)}={\left({x},{y},{2}{x}+{3}{y}\right)}={\left({1},{0},{2}\right)}{x}+{\left({0},{1},{3}\right)}{y}∈{S}{p}{a}{n}{\left\lbrace{\left({1},{0},{2}\right)},{\left({0},{1},{3}\right)}\right\rbrace}$$
It follows that the vectors (1,0,2), (0,1,3) spans the plane P1.

### Relevant Questions

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