(a) Find a vector parallel to the line of intersection of...

(a) Find a vector parallel to the line of intersection of the planes $2 x-y-3 z=0$ and $x+y+z=1$.
(b) Show that the point (1,-1,1) lies on both planes.
(c) Find parametric equations for the line of intersection.

Key Concepts

Parametric Equations for a Line

Parametric equations express the coordinates of points on a line as functions of a parameter, typically denoted by 't'. In the context of the intersection of two planes, once a point on the line and a direction vector (obtained from the cross product of the planes' normals) are known, parametric equations can be formulated to represent every point along the intersection line. This representation is versatile and widely used in both geometry and physics.

Verification of a Point on a Plane

To verify that a specific point lies on a plane, one substitutes the coordinates of the point into the plane's equation. If the resulting expression holds true, the point is on the plane. This method is a straightforward application of the definition of a plane and is commonly used to confirm the validity of points in geometric problems.

Plane Equation and Normal Vector

A plane in three-dimensional space is typically defined by an equation of the form ax + by + cz = d. The coefficients (a, b, c) form a normal vector to the plane, which is perpendicular to every line on the plane. This concept is fundamental for understanding how planes relate to one another in space, especially when examining intersections or angles between them.

Cross Product

The cross product is an operation on two vectors in three-dimensional space that produces a third vector perpendicular to both. In the context of intersecting planes, the cross product of the normal vectors of the two planes yields a direction vector that is parallel to the line where the planes intersect. This method is essential for finding the direction of the intersection line between two non-parallel planes.

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