Let `vecx, vecy` and `vecz` be three vectors each of magnitude `sqrt(2)` and the angle between each pair of them is `pi/3`. If `veca` is a non-zero vector perpendicular to `vecx` and `vecyxxvecz` and `vecb` is a non zero vector perpendicular to `vecy` and `veczxxvecx` then

Options

A. `vecb=(vecb.vecz)(vecz-vecx)`
B. `veca=(veca.vecy)(vecy-vecz)`
C. `veca.vecb=-(veca.vecy)(vecb.vecz)`
D. `veca=(veca.vecy)(vecz-vecy)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understand the Given Information
We have three vectors \( \vec{x}, \vec{y}, \vec{z} \) each with a magnitude of \( \sqrt{2} \) and the angle between each pair of them is \( \frac{\pi}{3} \).

### Step 2: Calculate the Dot Products
The dot product of two vectors can be calculated using the formula:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \]
where \( \theta \) is the angle between the vectors.

1. **Calculate \( \vec{x} \cdot \vec{y} \)**:
\[ \vec{x} \cdot \vec{y} = |\vec{x}| |\vec{y}| \cos\left(\frac{\pi}{3}\right) = \sqrt{2} \cdot \sqrt{2} \cdot \frac{1}{2} = 2 \cdot \frac{1}{2} = 1 \]

2. **Calculate \( \vec{y} \cdot \vec{z} \)**:
\[ \vec{y} \cdot \vec{z} = |\vec{y}| |\vec{z}| \cos\left(\frac{\pi}{3}\right) = \sqrt{2} \cdot \sqrt{2} \cdot \frac{1}{2} = 1 \]

3. **Calculate \( \vec{z} \cdot \vec{x} \)**:
\[ \vec{z} \cdot \vec{x} = |\vec{z}| |\vec{x}| \cos\left(\frac{\pi}{3}\right) = \sqrt{2} \cdot \sqrt{2} \cdot \frac{1}{2} = 1 \]

### Step 3: Establish the Perpendicular Vectors
Let \( \vec{a} \) be a vector perpendicular to \( \vec{x} \) and \( \vec{y} \times \vec{z} \). Let \( \vec{b} \) be a vector perpendicular to \( \vec{y} \) and \( \vec{z} \times \vec{x} \).

### Step 4: Find the Dot Product \( \vec{a} \cdot \vec{b} \)
Since \( \vec{a} \) is perpendicular to \( \vec{x} \) and \( \vec{y} \times \vec{z} \), and \( \vec{b} \) is perpendicular to \( \vec{y} \) and \( \vec{z} \times \vec{x} \), we can use the property of dot products for perpendicular vectors.

1. **Using the property of dot products**:
\[ \vec{a} \cdot \vec{b} = 0 \]
This is because \( \vec{a} \) and \( \vec{b} \) are perpendicular to vectors that are not parallel, thus their dot product is zero.

### Final Result
Thus, the value of \( \vec{a} \cdot \vec{b} \) is:
\[ \vec{a} \cdot \vec{b} = 0 \]