Let `veca=hati+2hatj+3hati, vecb=hati-hatj+2hatk and vecc=(x-2)hati-(x-3)hatj-hatk`. If `vecc` lies in the plane of `veca and vecb`, then `(1)/(x)` is equal to

Answer

To solve the problem, we need to determine the value of \( \frac{1}{x} \) given that the vector \( \vec{c} \) lies in the plane of vectors \( \vec{a} \) and \( \vec{b} \). This means that the vectors \( \vec{a}, \vec{b}, \vec{c} \) are coplanar, which can be determined using the scalar triple product. The scalar triple product of three vectors is zero if they are coplanar.

### Step-by-Step Solution:

1. **Define the Vectors**:
- \( \vec{a} = \hat{i} + 2\hat{j} + 3\hat{k} \)
- \( \vec{b} = \hat{i} - \hat{j} + 2\hat{k} \)
- \( \vec{c} = (x - 2)\hat{i} - (x - 3)\hat{j} - \hat{k} \)

2. **Write the Vectors in Component Form**:
- \( \vec{a} = (1, 2, 3) \)
- \( \vec{b} = (1, -1, 2) \)
- \( \vec{c} = (x - 2, -(x - 3), -1) = (x - 2, -x + 3, -1) \)

3. **Set Up the Determinant**:
To check if the vectors are coplanar, we need to compute the determinant of the matrix formed by these vectors:
\[ \begin{vmatrix} 1 & 2 & 3 \\ 1 & -1 & 2 \\ x - 2 & -x + 3 & -1 \end{vmatrix} = 0 \]

4. **Calculate the Determinant**:
Using the determinant formula:
\[ = 1 \begin{vmatrix} -1 & 2 \\ -x + 3 & -1 \end{vmatrix} - 2 \begin{vmatrix} 1 & 2 \\ x - 2 & -1 \end{vmatrix} + 3 \begin{vmatrix} 1 & -1 \\ x - 2 & -x + 3 \end{vmatrix} \]

- Calculate each of these 2x2 determinants:
- \( \begin{vmatrix} -1 & 2 \\ -x + 3 & -1 \end{vmatrix} = (-1)(-1) - (2)(-x + 3) = 1 + 2x - 6 = 2x - 5 \)

- \( \begin{vmatrix} 1 & 2 \\ x - 2 & -1 \end{vmatrix} = (1)(-1) - (2)(x - 2) = -1 - 2x + 4 = -2x + 3 \)

- \( \begin{vmatrix} 1 & -1 \\ x - 2 & -x + 3 \end{vmatrix} = (1)(-x + 3) - (-1)(x - 2) = -x + 3 + x - 2 = 1 \)

5. **Combine the Results**:
Substitute back into the determinant:
\[ 1(2x - 5) - 2(-2x + 3) + 3(1) = 0 \]
Simplifying this gives:
\[ 2x - 5 + 4x - 6 + 3 = 0 \]
\[ 6x - 8 = 0 \]

6. **Solve for \( x \)**:
\[ 6x = 8 \implies x = \frac{8}{6} = \frac{4}{3} \]

7. **Find \( \frac{1}{x} \)**:
\[ \frac{1}{x} = \frac{1}{\frac{4}{3}} = \frac{3}{4} \]

### Final Answer:
\[ \frac{1}{x} = \frac{3}{4} \]