Equation of the projection of the line 8 x-y-7 z=8, x +y+z=1...

Equation of the projection of the line $8 x-y-7 z=8, x$ $+y+z=1$ on the plane $5 x-4 y-z=5$ is
(A) $\frac{x-1}{1}=\frac{y}{2}=\frac{z}{-3}$
(B) $\frac{x}{1}=\frac{y-1}{2}=\frac{z}{-3}$
(C) $\frac{x}{1}=\frac{y}{2}=\frac{z-1}{-3}$
(D) $\frac{x}{1}=\frac{y+1}{-2}=\frac{z+1}{3}$

Key Concepts

Line in Three-Dimensional Space

A line in three dimensions can be represented using a point and a direction vector or as the intersection of two nonparallel planes. This representation is fundamental in analytic geometry for describing the set of points that form the line and for facilitating operations like projection and intersection with other geometric objects.

Plane in Three-Dimensional Space

A plane is defined by a linear equation in three variables and represents a flat, two-dimensional surface that extends infinitely in all directions within the space. The coefficients of the variables in the plane's equation correspond to the components of the plane’s normal vector, which plays a key role in determining perpendicular directions.

Projection of a Point onto a Plane

Projecting a point onto a plane involves finding the foot of the perpendicular from the point to the plane. This is done by utilizing the normal vector of the plane and ensures that the distance between the point and its projection is the shortest possible distance, which is crucial for applications such as finding shadows or solving optimization problems in geometry.

Orthogonal Projection of a Line onto a Plane

The orthogonal projection of a line onto a plane is constructed by projecting every point on the line onto the plane along lines that are perpendicular to the plane. This results in a new line that lies entirely within the plane, effectively capturing the 'shadow' of the original line when light is cast orthogonally to the plane. Understanding this concept is essential when translating three-dimensional configurations into two-dimensional representations.

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