Suppose I have a free vector $ \vec{w}$ and I have a plane $ P$ described the following way:

$$ \vec{r} = \vec{r_o} + a \vec{u} + b \vec{v}$$

Where $a,b$ are parameters to vary and $ \vec{u}$ and $ \vec{v}$ are vectors in the plane and $ \vec{r_o}$ is position vector to some vector in the plane

Suppose I wish to rotate the component of $ \vec{w}$ in the plane $P$ along an axis parallel to the normal of $P$, how would I write out the rotated new vector $ \vec{w'}$ which has the same component as $ w$ perpendicular to plane and the parallel part to plane as rotated?

I know to start I'd have to split up $ \vec{w}$ into components perpendicular and parallel to plane as follows;

$$ \vec{w} = \vec{w}_{\parallel} + \vec{w}_{\perp}$$

Not sure what I do after this

Visual depiction:

enter image description here


Black=original vector

Orange= vector part parallel to plane

Green= vector part parallel to plane which is rotated

Red= the new vector with the same perpendicular component by parallel part along plane rotated

  • $\begingroup$ $\vec v$ is already in the plane, so I think you are confused about something. Make a figure to explain what you want $\endgroup$ – Andrei Oct 11 '20 at 15:34
  • $\begingroup$ haha accidently used the same letter for everything. Will try to make a figure $\endgroup$ – Buraian Oct 11 '20 at 15:35

Given $\vec w_\parallel$ and $\vec w_\perp,$ let

$$\hat n = \frac{1}{\|\vec{w}_\perp\|} \vec{w}_\perp. $$

Then $\hat n$ is a unit normal vector to the plane. Further, let

$$ \vec w_= = \hat n \times \vec w_\parallel.$$

(The subscript $=$ here has no particular significance except that it looks somewhat like $\parallel$ rotated ninety degrees.) Then since $\hat n$ and $\vec w_\parallel$ are orthogonal, $\vec w_=$ is a vector in the plane $P$ orthogonal to $\vec w_\parallel$. Further, since $\hat n$ is a unit vector, $\vec w_=$ has the same magnitude as $\vec w_\parallel$.

Now to rotate $\vec w$ by angle $\theta$ around an axis perpendicular to $P,$ let

$$ \vec w' = \vec w_\perp + (\cos \theta)\vec w_\parallel + (\sin\theta)\vec w_=.$$

  • $\begingroup$ Wait so could you reconstruct the plane from this ? $\endgroup$ – Buraian Oct 11 '20 at 16:49
  • $\begingroup$ If by "reconstruct" you mean you could use $\vec w_\parallel$ and $\vec w_=$ in place of $\vec u$ and $\vec v$ when describing $P$, yes. $\endgroup$ – David K Oct 11 '20 at 17:02
  • $\begingroup$ Couldn't you have done this without defining w_{=} like wouldn't all this be possible with just $w_{ll}$ $\endgroup$ – Buraian Feb 16 at 12:16
  • $\begingroup$ Sure, just write $\hat n \times \vec w_\parallel$ instead of $\vec w_=.$ You don't really need a new symbol. $\endgroup$ – David K Feb 17 at 0:39

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