If `veca, vecb, vecc` are unit vectors such that `veca. vecb =0 = veca.vecc` and the angle between `vecb and vecc is pi/3` , then find the value of `|veca xx vecb -veca xx vecc|`

Explanation

To solve the problem step by step, we will use the properties of vectors, particularly focusing on the given conditions and the geometric interpretation of the cross product.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- Let \(\vec{a}, \vec{b}, \vec{c}\) be unit vectors.
- We know that \(\vec{a} \cdot \vec{b} = 0\) and \(\vec{a} \cdot \vec{c} = 0\). This implies that \(\vec{a}\) is perpendicular to both \(\vec{b}\) and \(\vec{c}\).
- The angle between \(\vec{b}\) and \(\vec{c}\) is \(\frac{\pi}{3}\).

2. **Express the Cross Products:**
- We need to find \(|\vec{a} \times \vec{b} - \vec{a} \times \vec{c}|\).
- We can factor out \(\vec{a}\) from the expression:
\[ \vec{a} \times \vec{b} - \vec{a} \times \vec{c} = \vec{a} \times (\vec{b} - \vec{c}). \]

3. **Magnitude of the Cross Product:**
- The magnitude of a cross product is given by:
\[ |\vec{a} \times (\vec{b} - \vec{c})| = |\vec{a}| |\vec{b} - \vec{c}| \sin \theta, \]
where \(\theta\) is the angle between \(\vec{a}\) and \(\vec{b} - \vec{c}\).
- Since \(\vec{a}\) is a unit vector, \(|\vec{a}| = 1\).

4. **Finding \(|\vec{b} - \vec{c}|\):**
- To find \(|\vec{b} - \vec{c}|\), we use the formula:
\[ |\vec{b} - \vec{c}| = \sqrt{|\vec{b}|^2 + |\vec{c}|^2 - 2 \vec{b} \cdot \vec{c}}. \]
- Since both \(\vec{b}\) and \(\vec{c}\) are unit vectors, \(|\vec{b}| = 1\) and \(|\vec{c}| = 1\).
- The dot product \(\vec{b} \cdot \vec{c} = |\vec{b}| |\vec{c}| \cos(\theta) = 1 \cdot 1 \cdot \cos\left(\frac{\pi}{3}\right) = \frac{1}{2}\).

5. **Substituting Values:**
- Now substituting into the formula:
\[ |\vec{b} - \vec{c}| = \sqrt{1^2 + 1^2 - 2 \cdot \frac{1}{2}} = \sqrt{1 + 1 - 1} = \sqrt{1} = 1. \]

6. **Finding the Final Magnitude:**
- Since \(\theta\) between \(\vec{a}\) and \(\vec{b} - \vec{c}\) is \(90^\circ\) (as \(\vec{a}\) is perpendicular to both \(\vec{b}\) and \(\vec{c}\)), \(\sin(90^\circ) = 1\).
- Therefore:
\[ |\vec{a} \times (\vec{b} - \vec{c})| = 1 \cdot 1 \cdot 1 = 1. \]

### Final Answer:
The value of \(|\vec{a} \times \vec{b} - \vec{a} \times \vec{c}|\) is \(1\).

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