Find the magnitude of the required vector using the given information.

Question

If $\vec a, \vec b, \vec c$ are non coplanar vectors such that $\vec b \times \vec c=\vec a$, $\vec c \times \vec a=\vec b$ and $\vec a \times \vec b=\vec c$, then find $|\vec a+\vec b+\vec c|$

I dont know how to begin the solution because no data regarding the magnitude or direction of vectors is given, and the answer is numeric. Can I get a hint to start the solution?

Answer 1

Hint:

• prove that $|a|=|b|=|c|$ (this might be a bit difficult)
• prove that $\vec{a},\vec{b},\vec{c}$ are mutually perpendicular
• from here deduce $|a|=|b|=|c|=1$
• now use $${|a+b+c|}^2=|a|^2+|b|^2+|c|^2+2a\cdot b+2b\cdot c+2a\cdot c$$

For the first step you may use the fact that

$$[a\space b\space c]=[b\space c\space a]=[c\space a\space b]$$ where $[a b c]$ dentes the scalar triple product of $\vec{a},\vec{b},\vec{c}$