Question

x = u + t1v + t2w
What conditions on the vectors u, v, w ∈ R3, would create an object that is not a plane?

Answer 1

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Answer on Question #76994 – Math – Linear Algebra

Question

$$x = u + t1v + t2w$$

What conditions on the vectors $u, v, w \in \mathbb{R}^3$, would create an object that is not a plane?

Solution

The expression $x = u + t1*v + t2*w$ sets a plane in $\mathbb{R}^3$ only if: 1) $u$ the radius-vector of a point; 2) $v, w$ are (nonzero) linear independent (not collinear) vectors.

If $u$ is the radius-vector of some point, then $x = u + t1*v + t2*w$ is not plane only if vectors $v$ and $w$ are linearly dependent ($v$ and $w$ are collinear).

Vectors $v, w \in \mathbb{R}^3$ must be linearly dependent. It means that $\exists a_1, a_2 \in \mathbb{R} \ (a_1 \neq 0 \text{ or } a_2 \neq 0) : a_1v + a_2w = 0$.