Linear Dependence

Two vectors in a plane are linearly dependent exactly when one is a scalar multiple of the other, equivalently when the determinant of their components vanishes. This determinant criterion is the standard JEE Advanced test for dependence and parallelism. Writing the vectors as rows (1, 2) and (2, t), the determinant is (1)(t) - (2)(2) = t - 4. Setting it to zero for dependence gives t - 4 = 0, hence t = 4. At this value the second vector equals exactly twice the first, since 2i + 4j = 2(i + 2j), confirming proportionality. Option 2 mistakes t for the leading coefficient rather than solving the determinant. Option -4 drops the sign while equating. Option 1 corresponds to setting t equal to the first component rather than enforcing proportional components. Linear dependence here means the two vectors are collinear, spanning only a line rather than the full plane. Plausibility check: substituting t = 4 makes the cross-multiplied ratios 1/2 = 2/4 equal, the hallmark of parallel vectors, so the dependence condition is genuinely met.