A mirror and a source of light are situated at the origin O...

A mirror and a source of light are situated at the origin $O$ and at a point on $O X$, respectively. A ray of light from the source strikes the mirror and is reflected. If the $D R s$ of the normal to the plane are $1,-1,1$, then $d$. $c$ 's of the reflected ray are (A) $\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$ (B) $-\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$ (C) $-\frac{1}{3},-\frac{2}{3},-\frac{2}{3}$ (D) $-\frac{1}{3},-\frac{2}{3}, \frac{2}{3}$

A mirror and a source of light are situated at the origin $O$ and at a point on $O X$, respectively. A ray of light from the source strikes the mirror and is reflected. If the $D R s$ of the normal to the plane are $1,-1,1$, then $d$. $c$ 's of the reflected ray are
(A) $\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$
(B) $-\frac{1}{3}, \frac{2}{3}, \frac{2}{3}$
(C) $-\frac{1}{3},-\frac{2}{3},-\frac{2}{3}$
(D) $-\frac{1}{3},-\frac{2}{3}, \frac{2}{3}$

Key Concepts

Law of Reflection

In optics, the law of reflection states that the angle at which a light ray strikes a surface (the angle of incidence) is equal to the angle at which the ray is reflected (the angle of reflection). In three-dimensional problems, this principle is applied relative to the normal of the surface, ensuring that the path of the incident ray, the normal, and the reflected ray all lie in the same plane.

Incident and Reflected Rays

These concepts involve the path of the incoming light ray from its source and the path of the ray after it reflects off the surface. In three-dimensional geometry, both the incident and reflected rays are represented as vectors whose directions can be characterized by their direction cosines.

Direction Cosines

Direction cosines are the cosines of the angles that a vector makes with the coordinate axes in three-dimensional space. They provide a way to describe the orientation of a vector and are crucial when determining the components of incident and reflected rays, ensuring that their magnitudes are normalized.

Normal Vector

The normal vector to a plane is a vector that is perpendicular to the plane. It is essential in reflection problems because it defines the orientation of the plane, and the reflection of any ray is computed with respect to this perpendicular direction.

Vector Reflection Formula

The vector reflection formula is an analytic expression that provides the reflected vector as a function of the incident vector and the normal vector. It is commonly written as r = i - 2*(i • n)*n, where i is the incident unit vector and n is the unit normal vector to the surface, ensuring that the magnitude of the reflected vector is preserved.

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