Perpendicular Vectors

Two vectors are perpendicular exactly when their dot product equals zero, since cos(90 degrees) = 0 forces the scalar product to vanish. Imposing the orthogonality condition and solving for the unknown is a frequent JEE Advanced parameter problem. Compute the dot product of (2, m, 1) and (1, -3, 2): (2)(1) + (m)(-3) + (1)(2) = 2 - 3m + 2 = 4 - 3m. Setting this equal to zero gives 4 - 3m = 0, so 3m = 4 and m = 4/3. Option 3/4 inverts the fraction by dividing incorrectly. Option -4/3 drops a sign when isolating m. Option 2 ignores the j-contribution entirely, solving only the constant terms. The positive value 4/3 is the unique scalar making the angle exactly 90 degrees. Geometrically, adjusting m tilts the first vector until its projection onto the second cancels to zero. Plausibility check: substituting m = 4/3 back gives 4 - 3(4/3) = 4 - 4 = 0, confirming the dot product vanishes and the vectors are indeed orthogonal.