Determine whether the following line is parallel to the plane 2 x-3 y+5 z=5: x=5+7 t, y=4+3 t,

step 1 So we have a line here and a line here x is 5 plus 7t y is 4 plus 3t z is negative 3 minus 2t. W

Determine whether the following line is parallel to the...

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Key Concepts

Parametric Representation of a Line

A line in space can be represented parametrically by specifying a point through which it passes and a direction vector. The direction vector indicates the line's orientation in space and is derived from the coefficients of the parameter in the line's equations.

Plane Equation and Normal Vector

A plane in three-dimensional space is often given by a linear equation in x, y, and z, from which a normal vector can be directly extracted. The normal vector is perpendicular to every vector lying within the plane, and its components correspond to the coefficients of x, y, and z in the plane’s equation.

Dot Product

The dot product is an operation that takes two vectors and returns a scalar, which is pivotal in determining the angle between the vectors. When the dot product of two vectors is zero, it indicates that the vectors are perpendicular (orthogonal) to each other.

Parallelism of a Line and a Plane

A line is parallel to a plane if its direction vector is perpendicular to the plane's normal vector. This condition is checked by computing the dot product of the line's direction vector and the plane's normal vector; if the result is zero, then the line is parallel to the plane.

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