Magnitude Of Resultant
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.
Select the correct option:
Solution
13
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.
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About This Question
- Subject
- mathematics
- Chapter
- vector algebra
- Topic
- magnitude of resultant
- Difficulty
- Medium
- Year
- 2025
Solution
Correct Answer:
13
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.
This medium difficulty mathematics question is from the chapter vector algebra, covering the topic of magnitude of resultant. It appeared in the 2025 exam.
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