Question

If the sum of the magnitude of two vectors is 32 and the magnitude of their resultant is 24. If the resultant is perpendicular to one of the vectors then what are the magnitudes of the two vectors?

Answer 1

Let A and B are two vectors .

a/c to question,

sum of magnitude of vectors = 32

|A| + |B| = 32 ......(1)

and resultant of A and B = 24

e.g., $\bf{\sqrt{|A|^2+|B|^2+2|A|.|B|cos\theta}}=24$

or, |A|² + |B|² + 2|A|.|B|.cos$\theta$ = 24² .......(2)

again, R is perpendicular on A. [you can assume B too]

we know angle made by resultant with one of the given vectors

tan90° = |B|sin$\theta$/(|A| + |B|cos$\theta$)

or, |A| + |B|cos$\theta$ = 0

or, cos$\theta$ = -|A|/|B|

now, from equation (2),

|A|² + |B|² + 2|A|.|B|. × -|A|/|B| = 24²

or, |B|² - |A|² = 24²

or, (|B| + |A|)(|B| - |A|) = 24²

from equation (1),

or, 32 × (|B| - |A|) = 576

or, (|B| - |A|) = 18 .......(3)

from equation (1) and (3),

|A| = 25 and |B| = 7