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  • The ratio of maximum and minimum magnitudes of the resultant of two vectors  and <img alt="\ve

The ratio of maximum and minimum magnitudes of the resultant of two vectors \vec{a} and \vec{b} is 3:1, Now \left | \vec{a} \right |=

Option: 1

| \vec{b}|


Option: 2

2| \vec{b}|


Option: 3

3| \vec{b}|


Option: 4

4| \vec{b}|


Answers (1)

best_answer

The resultant of two vectors is maximum when angle b/w them is 0 degree

Resultant is min when the angle is 180 degree

\begin{array}{l} \therefore \frac{|\vec{a}+\vec{b}|}{|\vec{a}-\vec{b}|}=\sqrt{\frac{a^{2}+b^{2}+2 a b}{a^{2}+b^{2}-2 a b}}=\frac{3}{1} \\ \Rightarrow a^{2}+b^{2}+2 a b=9 a^{2}+9 b^{2}-18 a b \\ \Rightarrow 8 a^{2}+8 b^{2}-20 a b=0 \\ \Rightarrow 2 a^{2}+2 b^{2}-5 a b=0 \end{array}

\begin{array}{l} \text { Let } a=x \cdot b \\ \Rightarrow 2 x^{2} b^{2}+2 b^{2}-5 a b^{2}=0 \\ \Rightarrow 2 x^{2}+2-5 x=0 \\ \Rightarrow 2 x^{2}-5 x+2=0 \\ \Rightarrow 2 x^{2}-4 x-x+2=0 \\ \Rightarrow 2 x(x-2)-1(x-2)=0 \\ \Rightarrow(2 x-1)(x-2)=0 \\ \therefore x=2 \text { or } x=\frac{1}{2} \\ \Rightarrow a=2 b \end{array}

 

i.e | \vec{a}|=2| \vec{b}|

Posted by

Devendra Khairwa

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