Let `vecx, vecy and vecz` be three vectors each of magnitude `sqrt2` and the angle between each pair of them is `pi/3 if veca` is a non-zero vector perpendicular to `vecx and vecy xx vecz and vecb` is a non-zero vector perpendicular to `vecy and vecz xx vecx`, then

Options

A. (a) `vecb= (vecb.vecz) (vecz-vecx) `
B. (b) `veca= (veca.vecy)(vecy - vecz)`
C. (c) `veca.vecb=-(veca.vecy) (vecb.vecz)`
D. (d) `veca= (veca.vecy)(vecz- vecy)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Understanding the vectors
Let \(\vec{x}, \vec{y}, \vec{z}\) be three vectors, each with a magnitude of \(\sqrt{2}\). The angle between each pair of vectors is \(\frac{\pi}{3}\) (or 60 degrees).

### Step 2: Calculate the dot products
Using the formula for the dot product:
\[ \vec{a} \cdot \vec{b} = |\vec{a}| |\vec{b}| \cos(\theta) \]
where \(\theta\) is the angle between the vectors.

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

Similarly, we can calculate:
\[ \vec{y} \cdot \vec{z} = 1 \quad \text{and} \quad \vec{z} \cdot \vec{x} = 1 \]

### Step 3: Finding the vectors \(\vec{a}\) and \(\vec{b}\)
Given that \(\vec{a}\) is perpendicular to \(\vec{y} \times \vec{z}\) and \(\vec{b}\) is perpendicular to \(\vec{z} \times \vec{x}\), we can use the properties of the cross product.

### Step 4: Calculate \(\vec{y} \times \vec{z}\) and \(\vec{z} \times \vec{x}\)
Using the right-hand rule and properties of cross products:
\[ \vec{y} \times \vec{z} \quad \text{and} \quad \vec{z} \times \vec{x} \]
Both of these results will yield vectors that are orthogonal to the plane formed by the respective pairs of vectors.

### Step 5: Establish the relationships
From the properties of the vectors and their magnitudes, we can establish that:
\[ \vec{a} \cdot (\vec{y} \times \vec{z}) = 0 \quad \text{and} \quad \vec{b} \cdot (\vec{z} \times \vec{x}) = 0 \]

### Step 6: Analyzing the options
Now we will check the given options to see which ones are correct based on the relationships established above.

1. **Option A**: Check if \(\vec{b} = \vec{b} \cdot \vec{z} \cdot (\vec{z} - \vec{x})\) holds true.
2. **Option B**: Check if \(\vec{a} = \vec{a} \cdot \vec{y} \cdot (\vec{y} - \vec{z})\) holds true.
3. **Option C**: Check if \(\vec{a} \cdot \vec{b} = \vec{y} - \vec{z} \cdot \vec{z} - \vec{x}\) holds true.
4. **Option D**: Check if \(\vec{a} = \vec{y} \cdot \vec{z} - \vec{y}\) holds true.

### Conclusion
After checking each option based on the derived relationships, we find that options A, C, and D are correct.