Let a, b and c b the distinct non-negative numbers. If the vectors `a hat(i) + a hat(j) + c hat(k), hat(i) + hat(k), c hat(i) + c hat(j) + b hat(k)` lie on a plane, then which one of the following is correct?

Options

A. c is the arithmetic mean of a and b
B. c is the geometric mean of a and b
C. c is the harmonic mean of a and b
D. c is equal to zero

Answer

To solve the problem, we need to determine the condition under which the three given vectors lie on the same plane. The vectors are:

1. \( \mathbf{v_1} = a \hat{i} + a \hat{j} + c \hat{k} \)
2. \( \mathbf{v_2} = \hat{i} + \hat{k} \)
3. \( \mathbf{v_3} = c \hat{i} + c \hat{j} + b \hat{k} \)

### Step 1: Set up the vectors
We can express the vectors in component form:
- \( \mathbf{v_1} = (a, a, c) \)
- \( \mathbf{v_2} = (1, 0, 1) \)
- \( \mathbf{v_3} = (c, c, b) \)

### Step 2: Use the condition for coplanarity
The vectors are coplanar if the scalar triple product (or box product) of the vectors is zero. This can be computed using the determinant of a matrix formed by the vectors.

### Step 3: Form the determinant
We can form the determinant as follows:

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

### Step 4: Calculate the determinant
Calculating the determinant, we have:

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

Calculating each of the 2x2 determinants:

1. \( \begin{vmatrix} 0 & 1 \\ c & b \end{vmatrix} = 0 \cdot b - 1 \cdot c = -c \)

2. \( \begin{vmatrix} 1 & 1 \\ c & b \end{vmatrix} = 1 \cdot b - 1 \cdot c = b - c \)

3. \( \begin{vmatrix} 1 & 0 \\ c & c \end{vmatrix} = 1 \cdot c - 0 \cdot c = c \)

Substituting back into the determinant:

\[ = a(-c) - a(b - c) + c(c) \]
\[ = -ac - ab + ac + c^2 \]
\[ = -ab + c^2 \]

### Step 5: Set the determinant to zero
For the vectors to be coplanar, we set the determinant equal to zero:

\[ -c^2 + ab = 0 \]

### Step 6: Rearranging the equation
Rearranging gives us the condition:

\[ c^2 = ab \]

### Conclusion
The condition for the vectors to be coplanar is that \( c^2 = ab \). This indicates that \( c \) is the geometric mean of \( a \) and \( b \).

### Final Answer
The correct option is that \( c \) is the geometric mean of \( a \) and \( b \).

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