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Addition and Subtraction of Algebraic Expression

The resultant...

Question

The resultant of two vectors $F_{1}$ and $F_{2}$ is $R_{1}$ .If the direction of $F_{2}$ is reversed the, new resultant is $R_{2}$ .Show that ${R_{1}}^{2} + {R_{2}}^{2} = 2 ({F_{1}}^{2} + {F_{2}}^{2}) .$

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Solution

Consider the problem
According to question

Resultant of two vectors $F_{1}$ and $F_{2}$ is $R_{1}$
then ,
$R_{1}^{2} = F_{1}^{2} + F_{2}^{2} + 2 F_{1} F_{2}$ ---- (i)

If the direction of $F_{2}$ reversed
then,
$R_{2}^{2} = F_{1}^{2} + (- F_{2})^{2} + 2 F_{1} (- F_{2})$

$R_{2}^{2} = F_{1}^{2} + F_{2}^{2} - 2 F_{1} F_{2}$ ---- (ii)
On adding (i) and (ii)

$R_{1}^{2} + R_{2}^{2} = [F_{1}^{2} + F_{2}^{2} + 2 F_{1} F_{2}] + [F_{1}^{2} + F_{2}^{2} - 2 F_{1} F_{2}]$

$R_{1}^{2} + R_{2}^{2} = 2 (F_{1}^{2} + F_{2}^{2})$

Hence prove.

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The resultant of two vectors F1 and F2 is R1.If the direction of F2 is reversed the, new resultant is R2.Show that R12+R22=2(F12+F22).

The resultant of two vectors $F_{1}$ and $F_{2}$ is $R_{1}$ .If the direction of $F_{2}$ is reversed the, new resultant is $R_{2}$ .Show that ${R_{1}}^{2} + {R_{2}}^{2} = 2 ({F_{1}}^{2} + {F_{2}}^{2}) .$