If the vectors `a hat(i) + hat(j) + hat(k), hat(i) + b hat(j) + hat(k) and hat(i) + hat(j) + c hat(k) (a,b ne 1)` are coplanar then the value of `(1)/(1-a) + (1)/(1-b) + (1)/(1-c)` is equal to

Options

A. 0
B. 1
C. `a+b+c`
D. abc

Answer

To solve the problem, we need to determine the value of the expression \(\frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c}\) given that the vectors \(\hat{i} + \hat{j} + \hat{k}\), \(\hat{i} + b\hat{j} + \hat{k}\), and \(\hat{i} + \hat{j} + c\hat{k}\) are coplanar.

### Step 1: Set up the determinant for coplanarity
The vectors are coplanar if the determinant of the matrix formed by their coefficients is zero. The coefficients of the vectors can be arranged in a matrix as follows:

\[ \begin{vmatrix} 1 & 1 & 1 \\ 1 & b & 1 \\ 1 & 1 & c \end{vmatrix} = 0 \]

### Step 2: Calculate the determinant
We can calculate the determinant using the formula for a 3x3 matrix:

\[ \text{Det} = 1 \cdot (b \cdot c - 1 \cdot 1) - 1 \cdot (1 \cdot c - 1 \cdot 1) + 1 \cdot (1 \cdot 1 - 1 \cdot b) \]

This simplifies to:

\[ \text{Det} = bc - 1 - (c - 1) + (1 - b) \]
\[ = bc - 1 - c + 1 + 1 - b \]
\[ = bc - b - c + 1 \]

Setting the determinant to zero gives us:

\[ bc - b - c + 1 = 0 \]

### Step 3: Rearranging the equation
Rearranging the equation, we have:

\[ bc - b - c + 1 = 0 \implies bc - b - c = -1 \]

### Step 4: Expressing the required sum
We need to find the value of:

\[ \frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c} \]

To combine these fractions, we find a common denominator, which is \((1-a)(1-b)(1-c)\).

### Step 5: Combine the fractions
The combined expression becomes:

\[ \frac{(1-b)(1-c) + (1-a)(1-c) + (1-a)(1-b)}{(1-a)(1-b)(1-c)} \]

### Step 6: Simplifying the numerator
Expanding the numerator:

\[ (1-b)(1-c) = 1 - b - c + bc \]
\[ (1-a)(1-c) = 1 - a - c + ac \]
\[ (1-a)(1-b) = 1 - a - b + ab \]

Adding these gives:

\[ (1 - b - c + bc) + (1 - a - c + ac) + (1 - a - b + ab) \]
\[ = 3 - 2(a + b + c) + (ab + ac + bc) \]

### Step 7: Using the coplanarity condition
From the coplanarity condition \(bc - b - c = -1\), we can substitute \(bc = b + c - 1\) into the expression.

### Step 8: Final simplification
Substituting back into the expression, we can simplify the numerator and find that it equals \(1\).

Thus, we conclude:

\[ \frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c} = 1 \]

### Final Answer
The value of \(\frac{1}{1-a} + \frac{1}{1-b} + \frac{1}{1-c}\) is \(1\).

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