Rotating vectors in planes

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:

Legend:

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

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

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

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