I have worked through the following problem and I am not sure, why this can not be solved using a dot product:
Edit: Text of the problem:
Split the forces $F_1$ and $F_2$ into their components along the $u$ and the $v$ axis.
The Question is: What part of $F_2$ is along the $v$ axis ?
Given $$F_2=500N, \alpha_2=45°, \beta=70° $$
The problem can be solved using the law of sines: $$ \frac{F_2}{\sin\beta}=\frac{F_{2,v}}{\sin(180-\beta-\alpha_2)} $$ and gives you 483N .
However I first tried using:
$$ F_2\cdot\cos(\alpha_2) = 353.6N $$
This would be the procedure if the axis were in a $90°$ angle to each other.
I thought my wrong first result might be owed to the missing unitcircle in the background that is reasoning the classical mechanical projections of vectors:
$$ \vec{F}= |F|\cos(\alpha)\vec{u_x}+ |F|\sin(\alpha)\vec{u_y} $$
But then I thought about the dot product, using a unit vector of the $v$-axis:
$$|F_2||u_v|\cdot\cos(\alpha_2) = F_{2,v} $$
Where I project $F_2$ on the axis of $v$ without scaling this projection.
But this would give me my wrong result again.
Leading to my question:
What am I missing here: Why am I not able to use the last two methods to get to the correct result ?
My first uneducated guess:
If you split a Vector into two components along a $90°$ axis, you receive a right triangle, the two components are in a $90°$ angle to each other.
In my example, the force has two components which are not in a $90°$ angle to each other then we can not treat them this way. Same goes for the dot product since I only receive a projection shorter than the vector lying along the $v$ axis.
Am I right here?
Best Answer
The $u$ and the $v$ axis are not perpendicular then the component of $\vec{F_2}$ along $v$ according to the following sketch
with $F_2=\|\vec{F_2}\|$ is indeed given by
$$F_{2,v}=F_2\frac{\sin(180-\beta-\alpha_2)}{\sin\beta}$$
and, assuming $\hat u$ directed on the right
$$F_{2,u}=-F_2\frac{\sin \alpha_2}{\sin\beta}$$
in such way that
$$\vec{F_2}= F_{2,u} \;\hat u+F_{2,v} \;\hat v$$
This kind of decomposition is different from the projection of $\vec{F_2}$ along the direction $\hat v$ which is indeed given by
$$F^{\text{proj}}_{2,v}=\vec{F_2}\cdot \hat v = F_2 \cos \alpha_2$$