Final answer:
Vector c, which lies along the x-axis and satisfies the condition a·(b×c) = 0, could be both (1, 0, 0) and (-1, 0, 0) as these vectors would result in a cross product with b that is perpendicular to a.
Step-by-step explanation:
The given vectors are a = 3i - 2j + k and b = -i - 4j + 3k, and vector c lies along the x-axis and satisfies a·(b×c) = 0. This equation suggests that vector a is orthogonal to the cross product of vectors b and c. Since vector c lies along the x-axis, it can be represented as c = xi for some scalar x. Using the properties of the cross product and the dot product, the only requirement for a·(b×c) to be zero is that vector c does not alter the direction of b×c away from being orthogonal to a.
Vector b×anything along the x-axis will result in a vector that lies in the yz-plane, which is orthogonal to any vector along the x-axis. Hence, any scalar multiple of the unit vector along the x-axis will satisfy c. The possible answers presented, (1, 0, 0) and (-1, 0, 0), both satisfy these conditions, so both A and D are technically correct answers to what vector c could be.