Question

Find a vector in which two of the three direction angles are $\alpha ={75}^o$ and $\beta ={55}^o$.

• A. $v=\cos {55}^o\hat{i}+\cos {75}^o\hat{j}+\cos {39}^o\hat{k}$
• B. $v=\cos {75}^o\hat{i}+\cos {55}^o\hat{j}+\cos {39}^o\hat{k}$
• C. $v=\cos {55}^o\hat{i}+\cos {39}^o\hat{j}+\cos {75}^o\hat{k}$
• D. $v=\cos {75}^o\hat{i}+\cos {39}^o\hat{j}+\cos {55}^o\hat{k}$

Answer 1

Answer: (B)

$v=\cos {75}^o\hat{i}+\cos {55}^o\hat{j}+\cos {39}^o\hat{k}$

$\alpha ={75}^{\circ },\beta ={55}^{\circ }{\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma =1{\cos }^2{75}^{\circ }+{\cos }^2{55}^{\circ }+{\cos }^2\gamma =1.0676+.3249+{\cos }^2\gamma =1{\cos }^2\gamma =1-.3925=.6075\cos \gamma =.779\Rightarrow \gamma ={\cos }^{-1}\left(0779\right)={39}^{\circ }$

So the vector is

$v=\cos {75}^{\circ }i+\cos {55}^{\circ }j+\cos {39}^{\circ }k$

Option $B$ is correct.