Question

The wavefront of a lightbeam is given by the equation $x+2y+3z=c$,(where c is arbitary constant) the angle made by the direction of light with the y-axis is:

• A. ${cos}^{-1}\frac{1}{\sqrt{14}}$
• B. ${cos}^{-1}\frac{2}{\sqrt{14}}$
• C. ${sin}^{-1}\frac{1}{\sqrt{14}}$
• D. ${sin}^{-1}\frac{2}{\sqrt{14}}$

Answer 1

The correct option is B ${cos}^{-1}\frac{2}{\sqrt{14}}$

Given,

Direction of light, $\overrightarrow{n}=\hat{i}+2\hat{j}+3\hat{k}$

The angle made by the direction of light with y-axis is

$cos\beta =\hat{n}.\hat{j}$

$cos\beta =\frac{\overrightarrow{n}}{|\overrightarrow{n}|}.\hat{j}$

$cos\beta =\frac{(\hat{i}+2\hat{j}+3\hat{k}).(0\hat{i}+\hat{j}+0\hat{k})}{\sqrt{(1{)}^2+(2{)}^2+(3{)}^2}}=\frac{2}{\sqrt{14}}$

$\beta =co{s}^{-1}\frac{2}{\sqrt{14}}$

The correct option is B.