Answer to Question #158055 in Classical Mechanics for Light Fury

Question #158055

A force f̂ is expressed with respect to the basis ê1 and ê2 by equation 2(3 ê1 +4 ê2) N. If the directions of ê1 makes an angle of 30 degree with f̂, find the vector ê2.


1
Expert's answer
2021-01-26T13:16:14-0500

Given,

f=6e1^+8e2^f=6\hat{e_1}+8 \hat{e_2}

The direction e1^\hat{e_1} makes angle with =30=30^\circ

tan(30)=8e2^6e2^\Rightarrow \tan(30^\circ)=|\frac{8\hat{e_2}}{6\hat{e_2}}|

13=8e2^6e1^\Rightarrow |\frac{1}{\sqrt{3}}|=|\frac{8\hat{e_2}}{6\hat{e_1}}|

squaring both side,

13=64e22^36e^2\Rightarrow \frac{1}{3}=\frac{64\hat{e_2^2}}{36\hat{e}^2}

36e12^=3×64e22^\Rightarrow 36 \hat{e_1^2}=3\times64\hat{e_2^2}

e12^=163e22^\Rightarrow \hat{e_1^2}=\frac{16}{3}\hat{e_2^2}


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