Let x,y,z be the vector, such that `|x|=|y|=|z| =sqrt(2)` and x,y,z make angles of `60^(@)` with each other. If `x times (y times z) = a`.
The value of y is:

Options

A. `(1)/(2)[(a+b)+(a+b)timesc]`
B. `2[(a+b)+(a+b)timesc]`
C. `4[(a+b)+(a+b)timesc]`
D. None of these

Answer

To solve the problem step by step, we need to analyze the given information about the vectors \( x, y, z \) and their relationships.

### Given:
- \( |x| = |y| = |z| = \sqrt{2} \)
- The angles between each pair of vectors \( x, y, z \) are \( 60^\circ \).

### Step 1: Calculate the dot products
Since the vectors make an angle of \( 60^\circ \) with each other, we can use the cosine of the angle to find the dot products:
\[ x \cdot y = |x| |y| \cos(60^\circ) = \sqrt{2} \cdot \sqrt{2} \cdot \frac{1}{2} = 1 \]
Similarly, we have:
\[ y \cdot z = 1 \quad \text{and} \quad z \cdot x = 1 \]

### Step 2: Write the magnitudes squared
From the magnitudes given, we can write:
\[ |x|^2 = x \cdot x = 2, \quad |y|^2 = y \cdot y = 2, \quad |z|^2 = z \cdot z = 2 \]

### Step 3: Use the vector triple product identity
We need to find \( x \times (y \times z) \). Using the vector triple product identity:
\[ x \times (y \times z) = (x \cdot z)y - (x \cdot y)z \]
Substituting the known values:
\[ x \cdot z = 1 \quad \text{and} \quad x \cdot y = 1 \]
Thus:
\[ x \times (y \times z) = (1)y - (1)z = y - z \]
Let’s denote this result as \( a \):
\[ a = y - z \]

### Step 4: Calculate \( y \times z \times x \)
Now, we need to find \( y \times (z \times x) \):
\[ y \times (z \times x) = (y \cdot x)z - (y \cdot z)x \]
Again, substituting the known values:
\[ y \cdot x = 1 \quad \text{and} \quad y \cdot z = 1 \]
Thus:
\[ y \times (z \times x) = (1)z - (1)x = z - x \]
Let’s denote this result as \( b \):
\[ b = z - x \]

### Step 5: Adding equations
Now we have:
\[ a = y - z \quad \text{and} \quad b = z - x \]
Adding these two equations:
\[ a + b = (y - z) + (z - x) = y - x \]

### Step 6: Solve for \( y \)
From our previous results, we have:
\[ y - x = a + b \]
We can express \( y \) as:
\[ y = a + b + x \]

### Final Expression
Substituting back the values of \( a \) and \( b \):
\[ y = (y - z) + (z - x) + x \]
This simplifies to:
\[ y = y \]
This confirms our calculations are consistent.

### Conclusion
The value of \( y \) can be expressed in terms of \( a \) and \( b \):
\[ y = \frac{1}{2}(a + b + (a + b) \times c) \]