Determine whether the given line is parallel to, intersects,...

Determine whether the given line is parallel to, intersects, or lies in the given plane. If the line is parallel to the plane, calculate its distance from the plane. If the line intersects the plane, find the point and angle at which they intersect. $$\mathbf{r}(t)=\langle t,-5-2 t,-1+3 t\rangle \text { and } 4 x-5 y+2 z=5$$

Determine whether the given line is parallel to, intersects, or lies in the given plane. If the line is parallel to the plane, calculate its distance from the plane. If the line intersects the plane, find the point and angle at which they intersect.
$$\mathbf{r}(t)=\langle t,-5-2 t,-1+3 t\rangle \text { and } 4 x-5 y+2 z=5$$

Key Concepts

Angle between a Line and a Plane

The angle of intersection between a line and a plane is defined as the angle between the line and its projection onto the plane. Alternatively, it is found by considering the complementary angle to that between the line’s direction vector and the plane’s normal vector. This calculation typically utilizes the dot product to relate the vectors and extract the appropriate angle.

Intersection of a Line and a Plane

To determine whether a line intersects a plane, one substitutes the parametric equations of the line into the plane's equation. This substitution results in an equation in one variable; if a solution exists, it provides the parameter value at the intersection point. If no solution exists, the line is parallel to the plane, and further analysis is needed to distinguish whether the line lies in the plane or is distinct.

Distance from a Line to a Plane

When a line is parallel to a plane, there is no intersection point, and the shortest distance between any point on the line and the plane remains constant. This distance can be calculated using the formula for the distance from a point to a plane, which involves the absolute value of the plane equation evaluated at the point, divided by the magnitude of the normal vector.

Parametric Representation of Lines

This concept involves expressing the equation of a line in terms of a parameter (often t), where the coordinates of any point on the line are given as functions of that parameter. The parametric form makes it straightforward to identify points on the line, its direction vector, and to facilitate substitution into other equations (such as those of planes) for analyzing relationships like intersection.

Plane Equation and Normal Vector

A plane in three-dimensional space is typically described by a linear equation in x, y, and z. The coefficients in this equation form a normal vector that is perpendicular to the plane. This normal vector is crucial in determining orientations, calculating angles of intersection with lines, and computing distances from points to the plane.

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