If `a=x""i+5j+7k, b=i+j-k, c=i+2j+2k` are coplanar then the value of x is

Answer

To determine the value of \( x \) such that the vectors \( \mathbf{a} = x \mathbf{i} + 5 \mathbf{j} + 7 \mathbf{k} \), \( \mathbf{b} = \mathbf{i} + \mathbf{j} - \mathbf{k} \), and \( \mathbf{c} = \mathbf{i} + 2 \mathbf{j} + 2 \mathbf{k} \) are coplanar, we will use the condition that the scalar triple product of the vectors must be zero. This can be expressed using the determinant of a matrix formed by the components of the vectors.

### Step-by-Step Solution:

1. **Set Up the Determinant**:
The vectors are given as:
\[ \mathbf{a} = x \mathbf{i} + 5 \mathbf{j} + 7 \mathbf{k} \]
\[ \mathbf{b} = 1 \mathbf{i} + 1 \mathbf{j} - 1 \mathbf{k} \]
\[ \mathbf{c} = 1 \mathbf{i} + 2 \mathbf{j} + 2 \mathbf{k} \]
We can form the determinant:
\[ \begin{vmatrix} x & 5 & 7 \\ 1 & 1 & -1 \\ 1 & 2 & 2 \end{vmatrix} = 0 \]

2. **Calculate the Determinant**:
We will calculate the determinant using the first row:
\[ = x \begin{vmatrix} 1 & -1 \\ 2 & 2 \end{vmatrix} - 5 \begin{vmatrix} 1 & -1 \\ 1 & 2 \end{vmatrix} + 7 \begin{vmatrix} 1 & 1 \\ 1 & 2 \end{vmatrix} \]

Now, we calculate each of the 2x2 determinants:
- For \( \begin{vmatrix} 1 & -1 \\ 2 & 2 \end{vmatrix} \):
\[ = (1)(2) - (-1)(2) = 2 + 2 = 4 \]
- For \( \begin{vmatrix} 1 & -1 \\ 1 & 2 \end{vmatrix} \):
\[ = (1)(2) - (-1)(1) = 2 + 1 = 3 \]
- For \( \begin{vmatrix} 1 & 1 \\ 1 & 2 \end{vmatrix} \):
\[ = (1)(2) - (1)(1) = 2 - 1 = 1 \]

Substituting back into the determinant:
\[ = x(4) - 5(3) + 7(1) = 4x - 15 + 7 \]
Simplifying:
\[ = 4x - 8 \]

3. **Set the Determinant to Zero**:
To find the value of \( x \):
\[ 4x - 8 = 0 \]

4. **Solve for \( x \)**:
\[ 4x = 8 \implies x = \frac{8}{4} = 2 \]

### Final Answer:
The value of \( x \) is \( 2 \).