How can I understand this vector-field?

Question

We are given a vectorfield $\vec{v}: \mathbb{R}^3 \rightarrow \mathbb{R}^3$ with $\vec{v}(\vec{x})= \|x\|^{\alpha} \vec{x}$ while $\alpha > 0$. Because I missed a lecture, I'm not sure if this is just a vector which has the length of $x$ to the power of $\alpha$ multiplied with every component, or if I'm missing out on some notation. Thanks for any help.

Answer 1

Multiplying a scalar by a vector will still keep it a vector (i.e. you can define a vector $\vec u$ as $\vec u= 3\vec x$)

Now $\|x\|$ is the "length" of the vector $x$, so it is a scalar (i.e. a number). So you can see $\vec v$ as a function that takes in a vector (for example $\vec x$) and gives you it multiplied by its norm to the power $\alpha$ that is a scalar ( i.e. it gives you $\|x\|^{\alpha}\vec x$)

If there still is a problem, please clarify it more.

Answer 2

Imagine your vectors as arrows in 3D space. At every $\vec x$, you're computing the length of $\vec x$ and stretching the arrow by a factor of $\|\vec x\|^\alpha$. As you get further away from the origin, your arrows are longer, so the scaling factor also grows, i.e. you're stretching them even more. So, if you're picturing the arrow $\vec x$, you should picture $v(\vec x)$ as a longer (or shorter, if $\|\vec x\|<1$) arrow put on top of $\vec x$.