A ray is incident on a plane surface. If î+ĵ-k̂ represents a vector along the direction of incid

step 1 Hello and welcome to this video solution of numerate. So here we are given that an incident arra

A ray is incident on a plane surface. If î+ĵ-k̂ represents a...

A ray is incident on a plane surface. If $\hat{i}+\hat{j}-\hat{k}$ represents a vector along the direction of incident ray. $\hat{i}+\hat{j}$ is a vector along normal on incident point in the plane of incident and reflected ray. Then vector along the direction of reflected ray is: (a) $-\frac{1}{\sqrt{19}}(-3 \hat{\mathbf{i}}+3 \hat{j}+\mathbf{k})$ (b) $\frac{1}{\sqrt{19}}(3 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\cdot \hat{\mathrm{k}})$ (c) $-\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$ (d) $\hat{\mathbf{k}}$

A ray is incident on a plane surface. If $\hat{i}+\hat{j}-\hat{k}$ represents a vector along the direction of incident ray. $\hat{i}+\hat{j}$ is a vector along normal on incident point in the plane of incident and reflected ray. Then vector along the direction of reflected ray is:
(a) $-\frac{1}{\sqrt{19}}(-3 \hat{\mathbf{i}}+3 \hat{j}+\mathbf{k})$
(b) $\frac{1}{\sqrt{19}}(3 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\cdot \hat{\mathrm{k}})$
(c) $-\frac{1}{\sqrt{3}}(\hat{i}+\hat{j}+\hat{k})$
(d) $\hat{\mathbf{k}}$

Key Concepts

Vector Reflection

This concept involves obtaining the reflected vector when an incident vector strikes a plane surface. The key idea is that the reflected vector is determined by subtracting twice the projection of the incident vector onto the unit normal of the plane. This ensures that the angle of incidence equals the angle of reflection relative to the normal, a principle that is fundamental in optics and physics.

Component Decomposition

This concept involves splitting a vector into two orthogonal components: one along the normal to the reflecting surface and one perpendicular to it. Decomposing the incident vector in this way allows for the isolation and manipulation of the normal component, which is then used to determine the change in direction after reflection.

Projection onto a Vector

Projection is the process of finding the component of one vector along the direction of another vector. In the context of reflections, projecting the incident vector onto the unit normal gives the magnitude of the component that gets reversed during reflection. This projection is essential to apply the vector reflection formula accurately.

Normalization of Vectors

Normalization refers to converting a vector into a unit vector by dividing it by its magnitude. Unit vectors are crucial in these calculations because they ensure that the projections and subsequent operations are scale-independent. Using normalized vectors simplifies the computation of angles and directions in reflection problems.

Please give Ace some feedback