If `vecr.veca=0, vecr.vecb=1and [vecr vecavecb]=1,veca.vecbne0,(veca.vecb)^(2)-|veca|^(2)|vecb|^(2)=1,` then find `vecr` in terms of `veca and vecb`.

Explanation

To solve the problem step by step, we will analyze the given conditions and derive the expression for the vector \( \vec{r} \) in terms of \( \vec{a} \) and \( \vec{b} \).

### Step 1: Understand the given conditions
We have the following conditions:
1. \( \vec{r} \cdot \vec{a} = 0 \) (This implies that \( \vec{r} \) is perpendicular to \( \vec{a} \))
2. \( \vec{r} \cdot \vec{b} = 1 \)
3. \( [\vec{r}, \vec{a}, \vec{b}] = 1 \) (This indicates that the scalar triple product is equal to 1)
4. \( \vec{a} \cdot \vec{b} \neq 0 \)
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