Finding the angle between unit vectors

Question

A vector $\vec{v}$ is called a unit vector if $\|\vec{v}\| = 1$.

Let $\vec{a},\vec{b}$, and $\vec{c}$ be unit vectors, such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$. Show that the angle between any two of these vectors is $120^\circ$. I have

$$ \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = -1 $$ $$ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} = -1 $$ $$ \vec{a} \cdot \vec{c} + \vec{b} \cdot \vec{c}= -1 $$

What should I do next?

Answer 1

Hints:

• Solve the system of linear equations to get the values of $a.b$, $b.c$, and $a.c$.
• The angle between $a$ and $b$ is $\cos^{-1}(a.b)$

Answer 2

I will continue on your approach:

$$ \vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = -1 $$ $$ \vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} = -1 $$ $$ \vec{a} \cdot \vec{c} + \vec{b} \cdot \vec{c}= -1 $$

If we subtract 2nd and 3rd equations we will have: $$\vec{a} \cdot \vec{b} - \vec{a} \cdot \vec{c} = 0,$$ Now we can add this new equation with your 1st equation: $$\vec{a} \cdot \vec{b} + \vec{a} \cdot \vec{c} = -1,$$

we will have: $$2\vec{a} \cdot \vec{b} = -1.$$

I believe it is easy now to conclude the angle between vectors $\vec{a}$ and $\vec{b}$. Similar approach will result with other angles...