Magnitude Of Resultant

Question

Two vectors of magnitudes 5 and 12 act at a point with an included angle of 90 degrees; evaluate the magnitude of their resultant vector.

RankGuru · Accepted Answer

The magnitude of the resultant of two vectors follows from the law |a + b|^2 = |a|^2 + |b|^2 + 2|a||b|cos(theta), derived by squaring the sum and applying the dot product. This parallelogram-law result is a foundational JEE Advanced computation for combining vectors. With |a| = 5, |b| = 12 and theta = 90 degrees, cos(90) = 0 eliminates the cross term, leaving |a + b|^2 = 25 + 144 + 0 = 169, so |a + b| = sqrt(169) = 13. Option 17 results from adding the magnitudes directly (5 + 12), valid only when the vectors are parallel. Option 7 subtracts the magnitudes (12 - 5), valid only for anti-parallel vectors. Option sqrt(119) wrongly uses a minus sign giving 144 - 25 instead of the sum. Because the vectors are perpendicular, the resultant is the hypotenuse of a right triangle with legs 5 and 12, the classic Pythagorean triple. Plausibility check: the resultant must lie between |12 - 5| = 7 and |12 + 5| = 17, and 13 falls within this interval, confirming the answer is admissible.

Magnitude Of Resultant

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