notation of vector multiplication formula

Question

I am looking at Rodrigues' rotation formula. It has been awhile since I took the relevant courses in school, and I'm having a little trouble with the notation.

According to wikipedia, the formula is as follows:

vrot = v cos θ + (k × v) sin θ + k (k · v) (1 - cos θ)

I am confused about two things: 1) Is v cos θ a scalar multiplication? 2) In the case of k (k · v) - let m = k · v. Is k (k · v) = k · m?

I think both come down to the same thing - I dont understand what product to calculate when no symbol is used, or how to determine which to use.

Answer 1

$k\cdot v$ is an inner product and returns a scalar, while $v \cos \theta$ is the vector $v$ being multiplied by the scalar $\cos \theta$. It makes no sense to say $k\cdot m$ as you have written because the inner product $\cdot$ is a binary operation on vectors. Instead you would write $mk$, which is the scalar $m$ multiplying the vector $k$.

Answer 2

Strictly speaking, the usual vector space axioms only define left-multiplication by a scalar, but it’s common to extend so that you can put the scalar anywhere in the term. If you like, whenever you see and “out of place” scalar $c$ like this, you could think of it as shorthand for $c$ times an appropriately-sized identity matrix.

It’s also fairly common practice to drop the parentheses around the arguments to trigonometric functions. When this is done, you will also usually see those factors come last in a term in order to disambiguate it. E.g., you have $\mathbf k\sin\theta$ instead of $\sin\theta\mathbf k$ or $\sin(\theta)\mathbf k$.

Finally, it’s generally the case that when mixing various vector and scalar products, the product of a scalar times anything will be denoted by simple juxtaposition, while the product of two vectors will have some binary operator symbol. Your proposed $\mathbf k\cdot m$ doesn’t make sense since the second operand of the dot product must be another vector, not a scalar.

Answer 3

Perhaps you'll find the French version of Wikipedia clearer than what you've found. I transcribe it here, adapting to your notations: $$\vec v_{\text{rot}}= (\cos\theta)\,\vec v+(1-\cos\theta\bigr)\bigl(\vec v\cdot\vec k)\,\vec k+(\sin\theta)\,\vec k\times\vec v$$