Two vectors $u$ and $v$ are given as $u = i-j+k$, $v = 2i-j+3k$;
I found the angle between the two vectors using $$\cos\theta= {(vu)\over(|v||u|)}$$ therefore simple dot-product, and the result is $$\cos\theta = {\sqrt(42)/7}$$.
Secondly, I found the vector orthogonal projection of u along direction v as $$u[orth.proj.] = {v(vu)\over(vv)}$$ = 1/7($6i-3j+9k$).
The length of such vector is then equal to |$u[orth.proj.]$| = 3/7 ($\sqrt(14)$).
I now quote the last question of the exercise:
Compute all vectors parallel to v whose orthogonal projection along the direction u has length 2.
For this, I thought of using the same formula I previously used to compute the angle between the two vectors :
$$\cos\theta = {(vu)\over(|v||u|)}$$
$$ |v|\cos\theta = {|v|(vu)\over(|v||u|) }$$
=$$(uv)\over|u|$$ which is exactly the length of v along u, therefore 2.
I also know that for a vector to be parallel to another, their cross-product must equal 0, but I have a hard time finding the answer; could you please help?
About projection, as mentioned, there is few to worry about. The definition follows below:
$\mathbf{Definition:}$ Given not-null vectors $u$ and $v$ on vector space V with inner product $\left<\cdot\,,\,\cdot\right>$ and induced norm $\lVert\cdot\rVert$ given by formula $\sqrt{\left<\cdot\,,\,\cdot\right>}$. The orthogonal projection of vector $v$ on vector $u$ is given by notation $\mbox{proj}_u v$ and formula $\frac{\left<u\,,\,v\right>}{\left<u\,,\,u\right>}\,u$.
$\mathbf{Proof:}$ In a geometrical example on the $R^2$ plane, the projection of vector $v$ on $u$ is given by product of projection norm, let us call $v_u$, equal to $\lVert v \rVert \cos{\theta}$, by a versor $\hat{v}$, equal to $\frac{u}{\lVert u \rVert}$. Since we know, the inner product equality $\left<u\,,\,v\right>=\lVert u \rVert \lVert v \rVert \cos{\theta}$, therefore, the projection norm $v_u$ is equal to $\frac{\left<u\,,\,v\right>}{\lVert u \rVert}$. It concludes the calculation by recognizing $\lVert u \rVert^2$ as $\left<u\,,\,u\right>$.
About parallel vectors to $v$, you multiply it by certain real constant $\alpha$, such that all vectors parallel to it corresponds to $\alpha \, v$.