Question

The resultant of two vectors at right angles is 5 N. If the angle between them is 120° and the resultant is \[\sqrt {13}\] then the vectors are ______.

Options

• \[\sqrt 3\] N, \[\sqrt 4\] N

• \[\sqrt 2\] N, \[\sqrt 5\] N

• 3 N, 4 N

• 7 N, 3 N

Answer 1

The resultant of two vectors at right angles is 5 N. If the angle between them is 120° and the resultant is \[\sqrt {13}\] then the vectors are 3 N, 4 N.

Explanation:

5 = \[\sqrt{\mathrm{F}_1^2+\mathrm{F}_2^2+2\mathrm{F}_1\mathrm{F}_2.\cos90^\circ}\]

25 = F₁² + F₂²   ...(i)

when θ = 120°

\[\sqrt {13}\] = \[\sqrt{\mathrm{F}_{1}^{2}+\mathrm{F}_{2}^{2}+2\mathrm{F}_{1}\mathrm{F}_{2}\cos120^{\circ}}\]

13 = 25 + 2F₁F₂ (-\[\frac {1}{2}\])

13 - 25 - F₁F₂

F₁F₂ = 12

F₂ = \[\frac {12}{F_1}\]   ...(ii)

Substituting equation (ii) in (i)

F₁² + \[\frac {144}{F_1^2}\] = 25

F₁⁴ +144 = 25 F₁²

F₁⁴ - 25 F₁² +144 = 0

(F₁² - 9) (F₁² - 16) = 0

F₁, F₂ = 3, 4