Let `vec(a) = hat(i) + 2hat(j) + 3hat(k) , vec(b) = 2hat(i) + hat(j) + hat(k), vec(c) = 3hat(i) + 2hat(j) + hat(k)` and `vec(d) = 3hat(i) - hat(j) - 2hat(k)`, then .
If `vec(a) xx (vec(b) xx vec(c)) = pvec(a) + qvec(b) + rvec(c)`, then find value of p,q are r.

Answer

To solve the problem, we need to find the values of \( p \), \( q \), and \( r \) in the equation:

\[ \vec{a} \times (\vec{b} \times \vec{c}) = p\vec{a} + q\vec{b} + r\vec{c} \]

where:
- \(\vec{a} = \hat{i} + 2\hat{j} + 3\hat{k}\)
- \(\vec{b} = 2\hat{i} + \hat{j} + \hat{k}\)
- \(\vec{c} = 3\hat{i} + 2\hat{j} + \hat{k}\)

### Step 1: Use the Vector Triple Product Identity

We can use the vector triple product identity:

\[ \vec{u} \times (\vec{v} \times \vec{w}) = (\vec{u} \cdot \vec{w})\vec{v} - (\vec{u} \cdot \vec{v})\vec{w} \]

In our case, let \(\vec{u} = \vec{a}\), \(\vec{v} = \vec{b}\), and \(\vec{w} = \vec{c}\). Thus, we have:

\[ \vec{a} \times (\vec{b} \times \vec{c}) = (\vec{a} \cdot \vec{c})\vec{b} - (\vec{a} \cdot \vec{b})\vec{c} \]

### Step 2: Calculate \( \vec{a} \cdot \vec{c} \)

Now we need to calculate the dot product \( \vec{a} \cdot \vec{c} \):

\[ \vec{a} \cdot \vec{c} = (1)(3) + (2)(2) + (3)(1) = 3 + 4 + 3 = 10 \]

### Step 3: Calculate \( \vec{a} \cdot \vec{b} \)

Next, we calculate the dot product \( \vec{a} \cdot \vec{b} \):

\[ \vec{a} \cdot \vec{b} = (1)(2) + (2)(1) + (3)(1) = 2 + 2 + 3 = 7 \]

### Step 4: Substitute Back into the Equation

Now we can substitute these values back into our equation:

\[ \vec{a} \times (\vec{b} \times \vec{c}) = 10\vec{b} - 7\vec{c} \]

This can be rewritten in the form:

\[ \vec{a} \times (\vec{b} \times \vec{c}) = 0\vec{a} + 10\vec{b} - 7\vec{c} \]

### Step 5: Identify the Coefficients

From the equation, we can identify the coefficients:

- \( p = 0 \)
- \( q = 10 \)
- \( r = -7 \)

### Final Values

Thus, the values are:

\[ p = 0, \quad q = 10, \quad r = -7 \]