The distance between the line (x-1)/(3)=(y+2)/(-2)=(z-1)/(2) and the plane 2 x+2 y-z-6=0 is (a)

step 1 Hello students in this question we have to determine the distance between the lines x minus 1 eq

The distance between the line (x-1)/(3)=(y+2)/(-2)=(z-1)/(2)...

Key Concepts

Parametric Representation of a Line in 3D

A line in three-dimensional space can be represented in a parametric or symmetric form which describes every point on the line using a parameter and a direction vector. This representation is fundamental in determining the position of points along the line and understanding its geometric relationship with other objects in space.

Plane Equation in 3D

A plane in three-dimensional space is typically defined by a linear equation of the form ax + by + cz + d = 0. This equation, which involves the plane's normal vector (a, b, c), characterizes the orientation of the plane and its position relative to the origin.

Distance from a Point to a Plane

The distance from a point in three-dimensional space to a plane is computed using the formula |ax0 + by0 + cz0 + d| divided by the square root of (a^2 + b^2 + c^2), where (x0, y0, z0) is the point in question. This formula is essential for finding the shortest distance between the point and the plane, a common task in vector geometry.

Parallelism Between a Line and a Plane

A line is parallel to a plane if the line’s direction vector is orthogonal to the normal vector of the plane. This orthogonality condition is checked using the dot product. When this condition is met, the distance between the line and the plane can be determined by calculating the distance from any point on the line to the plane.

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