Finding a sum of force vectors

Question

Three forces $\overrightarrow{F}= 2N\hat{i} -5N\hat{j},$ $ \overrightarrow{R}=10N(cos\theta\ \hat{i} + sin\theta\ \hat{j}),$ and $\overrightarrow{W}=WN\hat{j},$ with $W>0$ sum up to zero.

Determine $\theta$ and $W$ and draw the force vector $\overrightarrow{R}$ clearly showing its direction.

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So I have,

$\overrightarrow{0}=N(12\ cos\theta\ \hat{i} + 5W\ sin\theta\ \hat{j})$

I'm thinking maybe $W=\frac{12\ cos\theta}{5\ sin\theta}$

and plugging that in, I got

$0=12\ cos\theta + 12\ cos\theta = 24\ cos\theta$

...solve for $\theta$ but that doesn't feel right and I'm at a stand still.

Answer 1

HINT

We have that

• $x$ direction: $2+10\cos \theta=0$
• $y$ direction: $-5+10\sin \theta+W=0$

then solve for $\theta$ and $W$.

Note that "$N$" indicates Newtons and then we can not consider that in the calculation.