Vector Fields on $\mathbb R^3$

Question

The question I have to answer is: What is the set of points where two vector fields
$F,G$ in $\mathbb R^3$ are linearly dependent.

First, I cannot understand when it asks about points.In vector fields we have vectors, isn't it? So ,since they are linearly dependent can we say that all tangent vectors of $F$ are parallel to $G$'s tangent vectors?

Second , what is going on if I had exactly $3$ Linearly independent vector fields?

Is there a numerical example in order to understand vector fields and linear independence?

Answer 1

Vector fields $F$ and $G$ on $\mathbb{R}^3$ are maps $F,G:\mathbb{R}^3\to \mathbb{R}^3$. More precisely, a vector field assigns to each $p\in \mathbb{R}^3$ a vector in $\mathbb{R}^3$. So, denote $F_p$ as the vector $F(p)$ and similarly for $G$. The question is: given a pair of vector fields, at which $p\in \mathbb{R}^3$ are the fields linearly independent. That is, at which $p$ are $F_p$ and $G_p$ independent. A good way to do this is to calculate $$ F_p\times G_p$$ because $\lvert F_p\times G_p\rvert =0$ if and only $F_p=\lambda G_p$ if and only if $F_p$ and $G_p$ are linearly dependent. So, solving the equation $\lvert F_p\times G_p\rvert=0$ for $p\in \mathbb{R}^3$ will tell you at which $p\in \mathbb{R}^3$ the vector fields are linearly dependent.

Answer 2

A vector field on a manifold $M$ is a function $M\to TM$ that maps a point in $M$ to a vector in $TM$. In this case both $M$ and $TM$ are $\mathbb R^3$. And yes, we're talking about a tangent vector of $F$ that is parallel to a tangent vector of $G$ at the same point.

Let's pick $F: (x,y,z) \mapsto (x,y,z)$ and $G: (x,y,z)\mapsto (1,0,0)$.

Then the set of points where $F$ and $G$ are linearly dependent is: $$\{p\in\mathbb R^3\mid F(p)\text{ and }G(p)\text{ linearly dependent} \} = \{(x,0,0)\in\mathbb R^3\} = \mathbb R\times\{0\}\times\{0\}$$

An example of 3 vector fields that are pairwise linearly independent in every point, is for instance when each vector field maps to a different constant vector.