Question

Determine the basis for the set of vectors lying on the plane 𝑊 = {(𝑥, 𝑦, 𝑧): 3𝑥 + 𝑦 − 5𝑧 = 0}

Answer 1

$y = -3x + 5z,$ then the points that are satisfying the equation of the plane:

$$(x, -3x + 5z, z).$$

We need to find two different vectors satisfying above that are not a multiple of each other.

$$(x, -3x + 5z, z) = x(1, -3, 0) + z(0, 5, 1).$$

An obvious basis:

$$\begin{Bmatrix} (1, -3, 0), (0, 5, 1) \end{Bmatrix}.$$