Solving a Vector Problem with Cosine Law: Finding Magnitude and Angle Relations

[Santilopez10]

Homework Statement

The sum of 2 vectors $\vec u$ and $\vec v$ has length $|\vec u + \vec v|=10$, and its angle with one of the vectors is 35°, which` s lenght is 12. Find the lenght of the remaining vector and the angle between $\vec u$ and $\vec v$.

Relevant Equations

$$\langle \vec u,\vec v \rangle = |\vec u||\vec v|cos(\theta)$$

Okay, so the answer is quite easy if you draw a diagram and notice that cosine law solves everything rapidly. But at first, I tried doing some vector algebra and apply properties to see if I could get to something. This is what I could develop.

Consider $|\vec u|$=12, then $$\langle \vec u,(\vec u + \vec v) \rangle = 120 \cos(35º)$$
but $\langle \vec u,(\vec u + \vec v) \rangle = |\vec u|^2+\langle \vec u, \vec v \rangle$ so: $$\langle \vec u, \vec v \rangle =120 \cos(35º)-144$$

Now let's call the angle between $\vec u$ and $\vec v$ :$\phi$. Then the angle between $\vec v$ and $\vec u + \vec v$ = $\phi -35º$, and now we can get a system of equations for $\phi$ and $|\vec v|$:
1) $$ \langle \vec v,(\vec u + \vec v) \rangle = \langle \vec v,\vec u \rangle + |\vec v|^2= 10|\vec v|\cos(\phi-35º)$$
2) $$ \langle \vec u, \vec v \rangle = 12 |\vec v|\cos(\phi)$$
Maple returns $\phi$=2.3... (but correct answer is 123.6...) and $|\vec v|=6.8...$ which only $|\vec v|$ is correct (I had to use numeric solver). I know this is kind of overkill, but any approach to this problem would be great, thanks!

 

[HallsofIvy]

Certainly the cosine law works nicely. If you don't want to use it, set up a coordinate system so that vector u, with length 12 lies along the x-axis: u= <12, 0>. The sum of u and v, which has length 10 and is 35 degrees above u, so 35 degrees above the x-axis is u+ v= <10 cos(35), 10 sin(35)>. So v= <10 cos(35), 10 sin(35)>- u= <10 cos(35), 10 sin(35)>- <12, 0>= <10cos(35)- 12, 10sin(35_)>.