Question

If the projections of the line segment $AB$ on the coordinate axes are $2,3,6$, then the square of the sine of the angle made by $AB$ with $x=0$, is

• A. $\frac{3}{7}$
• B. $\frac{3}{49}$
• C. $\frac{4}{7}$
• D. $\frac{40}{49}$

Answer 1

The correct option is D $\frac{40}{49}$

Projections of the line segment $AB$ on coordinate axes are $2,3,$ and $6$
$\therefore$ the line can be $L:2\hat{i}+3\hat{j}+6\hat{k}$
Angle of line $L$ with $x=0i.e.\hat{j}$ is:

$\cos \theta =\frac{L\cdot \hat{j}}{|L||j|}=\frac{(2\hat{i}+3\hat{j}+6\hat{k})\cdot \hat{j}}{|(2\hat{i}+3\hat{j}+6\hat{k})||j|}=\frac{3}{7}$

${\sin }^2\theta =1-{\cos }^2\theta =1-{\left(\frac{3}{7}\right)}^2=\frac{40}{49}$