Vector Basics

Step 1: Understand the problem.
We are given two vector equations: We are tasked with finding the values of , , and that satisfy these conditions.

Step 2: Analyze the second equation.
The second equation can be simplified by factoring out : Since , , and are non-collinear, . Therefore, we must have:

Step 3: Substitute into the first equation.
Substituting into the first equation, we get: This simplifies to: Since and are non-collinear, the only solution to this equation is:

Step 4: Conclusion.
Thus, , , and , which corresponds to option (A).

Concept: Vectors are perpendicular if their dot product is zero.

Step 1: Express terms of G.P. Let G.P. be Taking log: Similarly for

Step 2: Form first vector:

Step 3: Dot product with second vector: Terms involving cancel: Remaining terms also cancel Dot product = 0 Final Answer: