Suppose you two forces ` vec Fand vec F` . How would you combine then in order to have resultant force of magnitudes (a) zero ` (b) 2 vec F and (c 0 vec F`.

Explanation

Ahe magnitude of resultant ` vec R` of the addition of two vectors ` vec A and vec B` is given by
` R sqrt ( A^2 + B^2 + 2 Abcos theta)`
where ` theta` is the angle between ` vec A and vec B`.
(a) If ` vec A = vec F, vec B = vec F` and ` R=0`, then
` =0 sqrt (F^@ + F^@ + 2 FF cos theta`
or ` 2 F^@ + 2 F^2 cos theta =0`
or ` cos theta =- 1 =- cos 180^@` or theta =180^@`
i.e. two vectors are acting in opposite derections.
(b) If ` vec A = vec F. vec B = vec F and R =2 F`, then (2 F)^@ = F^2 + F^2 + 2 FF cos theta`
or ` cos theta =1 or theta =0^@`
i.e., two vectors are acting in the same direction.
(c ) ` vec A = vec F. vec B= vec F` and R = F`, then
` F^2 = F^2 + F^2 + 2 FF cos theta =2 F^2 (1 - cos theta )`
or ` 1- cos theta =1/2 or cos theta =1 - 1/2/ =1/2 = cos 60^@`
i.e., two vectors are inclined at an angle of ` 60^@` .