- #1

Saitama

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**Problem:**

Let $\vec{a}$,$\vec{b}$ and $\vec{c}$ be non-coplanar unit vectors, equally incline to one another at an angle $\theta$. If $\vec{a}\times \vec{b} + \vec{b}\times \vec{c}=p\vec{a}+q\vec{b}+r\vec{c}$. Find scalars $p$,$q$ and $r$ in terms of $\theta$.

**Attempt:**

Taking the dot product on both sides successively with $\vec{a}$,$\vec{b}$ and $\vec{c}$, I get the following three equations:

$$\begin{array}

\\

[\vec{a} \vec{b} \vec{c}]=p+q\cos\theta+r\cos\theta \\

0=p\cos\theta+q+r\cos\theta \\

[\vec{a} \vec{b} \vec{c}]=p\cos\theta+q\cos\theta+r \\

\end{array}

$$

I am not sure how to proceed after this. I guess I have to express $[\vec{a} \vec{b} \vec{c}]$ in terms of $\theta$ but I don't see how.

Any help is appreciated. Thanks!

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