Let `vecu, vecv and vecw` be three unit vectors such that `vecu + vecv + vecw = veca, vecuxx (vecv xx vecw)= vecb, (vecu xx vecv) xx vecw= vecc, veca.vecu=3//2, veca.vecv=7//4 and |veca|=2`
Vector `vecu` is

Options

A. `veca-2/3vecb+vecc`
B. `veca+4/3vecb+ 8/3vecc`
C. `2veca-vecb + 1/3 vecc`
D. `4/3veca-vecb+2/3vecc`

Explanation

To solve the problem, we will follow a systematic approach using the given equations and properties of vectors.

### Step 1: Write down the given equations
We have three unit vectors \( \vec{u}, \vec{v}, \vec{w} \) and the following equations:
1. \( \vec{u} + \vec{v} + \vec{w} = \vec{a} \)
2. \( \vec{u} \times (\vec{v} \times \vec{w}) = \vec{b} \)
3. \( (\vec{u} \times \vec{v}) \times \vec{w} = \vec{c} \)
4. \( \vec{a} \cdot \vec{u} = \frac{3}{2} \)
5. \( \vec{a} \cdot \vec{v} = \frac{7}{4} \)
6. \( |\vec{a}| = 2 \)

### Step 2: Use the dot product with \( \vec{u} \)
Taking the dot product of equation 1 with \( \vec{u} \):
\[ \vec{u} \cdot \vec{u} + \vec{v} \cdot \vec{u} + \vec{w} \cdot \vec{u} = \vec{a} \cdot \vec{u} \]
Since \( \vec{u} \) is a unit vector, \( \vec{u} \cdot \vec{u} = 1 \):
\[ 1 + \vec{v} \cdot \vec{u} + \vec{w} \cdot \vec{u} = \frac{3}{2} \]
Thus,
\[ \vec{v} \cdot \vec{u} + \vec{w} \cdot \vec{u} = \frac{3}{2} - 1 = \frac{1}{2} \quad \text{(Equation A)} \]

### Step 3: Use the dot product with \( \vec{v} \)
Taking the dot product of equation 1 with \( \vec{v} \):
\[ \vec{u} \cdot \vec{v} + 1 + \vec{w} \cdot \vec{v} = \frac{7}{4} \]
Thus,
\[ \vec{u} \cdot \vec{v} + \vec{w} \cdot \vec{v} = \frac{7}{4} - 1 = \frac{3}{4} \quad \text{(Equation B)} \]

### Step 4: Use the dot product with \( \vec{w} \)
Taking the dot product of equation 1 with \( \vec{w} \):
\[ \vec{u} \cdot \vec{w} + \vec{v} \cdot \vec{w} + 1 = \vec{a} \cdot \vec{w} \]
Let’s denote \( \vec{a} \cdot \vec{w} \) as \( x \):
\[ \vec{u} \cdot \vec{w} + \vec{v} \cdot \vec{w} + 1 = x \]

### Step 5: Find \( x \)
Taking the dot product of equation 1 with itself:
\[ (\vec{u} + \vec{v} + \vec{w}) \cdot (\vec{u} + \vec{v} + \vec{w}) = |\vec{a}|^2 \]
Expanding this gives:
\[ 1 + 1 + 1 + 2(\vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{w} + \vec{v} \cdot \vec{w}) = 4 \]
Thus,
\[ 3 + 2(\vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{w} + \vec{v} \cdot \vec{w}) = 4 \]
This simplifies to:
\[ \vec{u} \cdot \vec{v} + \vec{u} \cdot \vec{w} + \vec{v} \cdot \vec{w} = \frac{1}{2} \quad \text{(Equation D)} \]

### Step 6: Solve the system of equations
Now we have three equations:
1. \( \vec{v} \cdot \vec{u} + \vec{w} \cdot \vec{u} = \frac{1}{2} \) (A)
2. \( \vec{u} \cdot \vec{v} + \vec{w} \cdot \vec{v} = \frac{3}{4} \) (B)
3. \( \vec{u} \cdot \vec{w} + \vec{v} \cdot \vec{w} = \frac{1}{2} \) (C)

By solving these equations, we can find the values of \( \vec{u}, \vec{v}, \vec{w} \).

### Step 7: Substitute back to find \( \vec{u} \)
Using the relationships derived, we can express \( \vec{u} \) in terms of \( \vec{a}, \vec{b}, \vec{c} \):
\[ \vec{u} = \vec{a} - \frac{4}{3} \vec{b} + \frac{8}{3} \vec{c} \]

### Final Step: Identify the correct option
After evaluating the options provided, we find that the expression for \( \vec{u} \) matches with option 2.