Question

If \( a \), \( b \), \( c \) are non-negative distinct numbers and \( ai + aj + ck \), \( i + j + k \), and \( ci + cj + bk \) are coplanar vectors, then

• A. \( a, c, b \) are in A.P.
• B. \( a, b, c \) are in G.P.
• C. \( a, c, b \) are in G.P.
• D. \( a, b, c \) are in A.P.

Hint:

For coplanar vectors, the scalar triple product must be zero, and this condition can be used to derive relationships between the coefficients.

Answer 1

The Correct Option is C

Step 1: Use the condition for coplanar vectors.
For three vectors to be coplanar, the scalar triple product must be zero. Use this condition to form an equation relating \( a \), \( b \), and \( c \).
Step 2: Solve the equation.
After solving the equation, we find that \( a, c, b \) must be in geometric progression (G.P.).
Step 3: Conclusion.
The correct answer is (C) \( a, c, b \) are in G.P..