If `hata, hatb and hatc` are non-coplanar unti vectors such that `[hata hatb hatc]=[hatb xx hatc" "hatc xx hata" "hata xx hatb]`, then find the projection of `hatb+hatc` on `hata xx hatb` .

Answer

To solve the problem, we need to find the projection of the vector \(\hat{b} + \hat{c}\) on the vector \(\hat{a} \times \hat{b}\).

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We are given three non-coplanar unit vectors \(\hat{a}\), \(\hat{b}\), and \(\hat{c}\). The notation \([\hat{a}, \hat{b}, \hat{c}]\) represents the scalar triple product, which gives the volume of the parallelepiped formed by these vectors. The condition given is:
\[ [\hat{a}, \hat{b}, \hat{c}] = [\hat{b} \times \hat{c}, \hat{c} \times \hat{a}, \hat{a} \times \hat{b}] \]

2. **Properties of Unit Vectors**:
Since \(\hat{a}\), \(\hat{b}\), and \(\hat{c}\) are unit vectors, we have:
\[ |\hat{a}| = |\hat{b}| = |\hat{c}| = 1 \]

3. **Volume Calculation**:
The volume \(V\) formed by the vectors can be expressed as:
\[ V = |\hat{a} \cdot (\hat{b} \times \hat{c})| \]
Since the vectors are non-coplanar, \(V\) is non-zero.

4. **Using the Scalar Triple Product**:
The scalar triple product can also be expressed as:
\[ V^2 = [\hat{a}, \hat{b}, \hat{c}]^2 = |\hat{a}|^2 |\hat{b}|^2 |\hat{c}|^2 - (\hat{a} \cdot \hat{b})^2 - (\hat{b} \cdot \hat{c})^2 - (\hat{c} \cdot \hat{a})^2 + 2(\hat{a} \cdot \hat{b})(\hat{b} \cdot \hat{c})(\hat{c} \cdot \hat{a}) \]
Given that the volume is non-zero, we conclude that the vectors are mutually perpendicular.

5. **Finding the Projection**:
The projection of a vector \(\mathbf{p}\) onto another vector \(\mathbf{q}\) is given by:
\[ \text{proj}_{\mathbf{q}} \mathbf{p} = \frac{\mathbf{p} \cdot \mathbf{q}}{|\mathbf{q}|^2} \mathbf{q} \]
Here, we need to find the projection of \(\hat{b} + \hat{c}\) on \(\hat{a} \times \hat{b}\).

6. **Calculating the Dot Product**:
We compute:
\[ (\hat{b} + \hat{c}) \cdot (\hat{a} \times \hat{b}) = \hat{b} \cdot (\hat{a} \times \hat{b}) + \hat{c} \cdot (\hat{a} \times \hat{b}) \]
The first term \(\hat{b} \cdot (\hat{a} \times \hat{b}) = 0\) because the dot product of any vector with itself is zero.

The second term can be simplified using the properties of the scalar triple product:
\[ \hat{c} \cdot (\hat{a} \times \hat{b}) = [\hat{a}, \hat{b}, \hat{c}] = V \]

7. **Magnitude of the Cross Product**:
The magnitude of \(\hat{a} \times \hat{b}\) is given by:
\[ |\hat{a} \times \hat{b}| = |\hat{a}| |\hat{b}| \sin(\theta) = 1 \cdot 1 \cdot 1 = 1 \quad (\text{since } \hat{a} \text{ and } \hat{b} \text{ are perpendicular}) \]

8. **Final Projection Calculation**:
Therefore, the projection of \(\hat{b} + \hat{c}\) on \(\hat{a} \times \hat{b}\) is:
\[ \text{proj}_{\hat{a} \times \hat{b}} (\hat{b} + \hat{c}) = \frac{V}{|\hat{a} \times \hat{b}|^2} (\hat{a} \times \hat{b}) = V \cdot (\hat{a} \times \hat{b}) \]
Since \(V = 1\), we conclude:
\[ \text{proj}_{\hat{a} \times \hat{b}} (\hat{b} + \hat{c}) = 1 \cdot (\hat{a} \times \hat{b}) = \hat{a} \times \hat{b} \]

### Conclusion:
The projection of \(\hat{b} + \hat{c}\) on \(\hat{a} \times \hat{b}\) is \(1\).