algebra-precalculusvectors
Let $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$ be unit vectors, such that $\mathbf{a}+\mathbf{b}+\mathbf{c} = \mathbf{0}$. Show that the angle between any two of these vectors is $120^\circ$.
Hi,
I have been having some trouble with this problem.
I have tried to assign variables to the vectors, and creating various equations with them. However, I can't figure out how to use the $\mathbf{a+b+c=0}$ information.
Thanks!
If the vectors form any closed polygon then traversing them all should bring you back where you started:
$$ \vec{a} + \vec{b} + \vec{c} = 0 $$
In this case the vectors do not form a triangle, though they do if you replace $\vec{b}$ by $-\vec{b}$ i.e. make $\vec{b}$ point in the other direction.
They are position vectors in $\mathbb{R}^3$. Two vectors describe a plane so I presume you are being asked to find the angle between the two vectors in the plane they describe.
I've added a small illustration (done in Geogebra) to show what I mean. The two points $A$ and $B$ are described by the vectors you gave. The blue plane is described by $A, B$ and the origin and the angle between these two vectors is $153^{\circ}$ (not sure if the angle is properly visible). You can compute this angle via the dot product. If you used the dot product and got to $157.6^{\circ}$ there may be a rounding error of some description in your calculation. Either way, I believe this may be what your teacher is looking for. Does this help?
Best Answer
The angle $\alpha$ betwwen unit vectors $\mathbf x,\mathbf y$ is defined by the property $\cos\alpha=\mathbf x\cdot\mathbf y$. So here we want to show that $\mathbf a\cdot\mathbf b=\mathbf a\cdot\mathbf c=\mathbf b\cdot\mathbf c=-\frac12$. By multiplying the given equation with $\mathbf a$, you get $$1+\mathbf a\cdot\mathbf b+\mathbf a\cdot \mathbf c=0 $$ If you do the same with mutltiplicatoin by $\mathbf b$ and multiplication by $\mathbf c$, you obtain three equations in the three unknowns $\mathbf a\cdot \mathbf b$, $\mathbf b\cdot \mathbf c$, $\mathbf a\cdot \mathbf c$ that you can solve.