A light wave is propagating with plane wave f | vedclass

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(A) The equation of the plane wave front is given by $x+y+z = C$.The normal vector to this plane is $\vec{n} = 1\hat{i} + 1\hat{j} + 1\hat{k}$.The dir

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$\cos ^{-1}\left(\frac{1}{\sqrt{3}}\right)$

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(A) The equation of the plane wave front is given by $x+y+z = C$.The normal vector to this plane is $\vec{n} = 1\hat{i} + 1\hat{j} + 1\hat{k}$.The direction of wave propagation is along the normal to the wave front,so the direction vector is $\vec{v} = \hat{i} + \hat{j} + \hat{k}$.The angle $\alpha$ made by the propagation vector with the $x$-axis is given by the formula for the direction cosine: $\cos \alpha = \frac{\vec{v} \cdot \hat{i}}{|\vec{v}| |\hat{i}|}$.Calculating the dot product: $\vec{v} \cdot \hat{i} = (1)(1) + (1)(0) + (1)(0) = 1$.Calculating the magnitude: $|\vec{v}| = \sqrt{1^2 + 1^2 + 1^2} = \sqrt{3}$.Therefore,$\cos \alpha = \frac{1}{\sqrt{3} \cdot 1} = \frac{1}{\sqrt{3}}$.Thus,$\alpha = \cos^{-1}\left(\frac{1}{\sqrt{3}}\right)$.

$A$ light wave is propagating with plane wave fronts of the type $x+y+z=$ constant. The angle made by the direction of wave propagation with the $x$-axis is

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