Question

The resultant of two vectors F1 and F2 is R1. If the direction of F2 is reversed, the new resultant
is R2. Show that (R1)² + (R2)² = 2[(F1)²+(F2)²].
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Answer 1

R2 is reversed , then R2 is (F1-F2 )

Answer 2

Answer:

$R_1^2+R_2^2=2[F_1^2+F_2^2]$

Explanation:

It is given that,

$\vec{R_1}=\vec{F_1}+\vec{F_2}$  (1)

The magnitude of equation (1) is given as,

$R_1^2=F_1^2+F_2^2+2F_1F_2cos\theta$     (2)

If F₂'s direction is reversed,

$\vec{R_2}=\vec{F_1}-\vec{F_2}$    (3)

The magnitude of equation (3) is given as,

$R_2^2=F_1^2+F_2^2-2F_1F_2cos\theta$    (4)

Adding equations (2) and (4) we get;

$R_1^2+R_2^2=F_1^2+F_2^2+2F_1F_2cos\theta+F_1^2+F_2^2-2F_1F_2cos\theta$

$R_1^2+R_2^2=2F_1^2+2F_2^2$

$R_1^2+R_2^2=2[F_1^2+F_2^2]$

Hence, proved.

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