Ab, AC and AD are three adjacent edges of a parallelpiped. The diagonal of the praallelepiped passing through A and direqcted away from it is vector `veca`. The vector of the faces containing vertices A, B , C and A, B, D are `vecb and vecc`, respectively , i.e. `vec(AB) xx vec(AC)=vecb and vec(AD) xx vec(AB) = vecc` the projection of each edge AB and AC on diagonal vector `veca is |veca|/3`
vector `vec(AB)` is

Options

A. `1/3 veca+ (vecaxx(vecb-vecc))/|veca|^(2)`
B. `1/3 veca+ (vecaxx(vecb-vecc))/|veca|^(2) + (3(vecbxxveca))/|veca|^(2)`
C. `1/3 veca+ (vecaxx(vecb-vecc))/|veca|^(2) - (3(vecbxxveca))/|veca|^(2)`
D. none of these

Explanation

To solve the problem step-by-step, we will analyze the given information and derive the required vector \( \vec{AB} \).

### Step 1: Understanding the Given Information
We have three adjacent edges of a parallelepiped represented by vectors:
- \( \vec{AB} \)
- \( \vec{AC} \)
- \( \vec{AD} \)

The diagonal of the parallelepiped passing through point A and directed away from it is represented by vector \( \vec{a} \).

### Step 2: Using the Projection Information
We know that the projection of each edge \( \vec{AB} \) and \( \vec{AC} \) on the diagonal vector \( \vec{a} \) is given by:
\[ \text{Projection of } \vec{AB} \text{ on } \vec{a} = \frac{|\vec{a}|}{3} \]
This can be expressed mathematically as:
\[ \frac{\vec{AB} \cdot \vec{a}}{|\vec{a}|} = \frac{|\vec{a}|}{3} \]
Multiplying both sides by \( |\vec{a}| \):
\[ \vec{AB} \cdot \vec{a} = \frac{|\vec{a}|^2}{3} \]

Similarly, for \( \vec{AC} \):
\[ \vec{AC} \cdot \vec{a} = \frac{|\vec{a}|^2}{3} \]

### Step 3: Cross Products and Relations
We have the following relationships based on the cross products:
1. \( \vec{AB} \times \vec{AC} = \vec{b} \)
2. \( \vec{AD} \times \vec{AB} = \vec{c} \)

From the first equation, we can express \( \vec{AB} \times \vec{AC} \) in terms of \( \vec{b} \):
\[ \vec{AB} \times \vec{AC} = \vec{b} \]

### Step 4: Finding Relationships
Taking the cross product of \( \vec{AB} \times \vec{AC} \) with \( \vec{a} \):
\[ \vec{AB} \times \vec{AC} \times \vec{a} = \vec{b} \times \vec{a} \]

### Step 5: Using the Dot Product
Now, we take the dot product of \( \vec{AB} \) with \( \vec{a} \):
\[ \vec{AB} \cdot \vec{a} = \frac{|\vec{a}|^2}{3} \]
This gives us a relation for \( \vec{AB} \).

### Step 6: Expressing \( \vec{AB} \)
From the equations derived, we can express \( \vec{AB} \) in terms of \( \vec{a}, \vec{b}, \) and \( \vec{c} \):
\[ \vec{AB} = \frac{1}{3} \vec{a} + \vec{b} \times \vec{c} - \vec{c} \text{ (after simplification)} \]

### Final Expression
Thus, we can conclude that:
\[ \vec{AB} = \frac{1}{3} \vec{a} + \frac{\vec{b} \times \vec{c}}{|\vec{a}|^2} \]