Question

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.

Answer 1

Given,

$f=6\hat{e_1}+8 \hat{e_2}$

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

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

$\Rightarrow |\frac{1}{\sqrt{3}}|=|\frac{8\hat{e_2}}{6\hat{e_1}}|$

squaring both side,

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

$\Rightarrow 36 \hat{e_1^2}=3\times64\hat{e_2^2}$

$\Rightarrow \hat{e_1^2}=\frac{16}{3}\hat{e_2^2}$