Lecture on Planes and Vectors

These flashcards cover key terms and concepts related to planes, vectors, and intersections, based on the provided lecture notes.

Vector Parametric Equation for a Plane

r = ro + sd + te, where ro is a point on the plane, d and e are direction vectors, and s, t are parameters.

Two Parameters for a Plane

A plane is 2-dimensional and requires two parameters for its representation.

Independent Vectors

Vectors d and e must be independent to define a plane, meaning one cannot be a scalar multiple of the other.

One Point and Two Vectors

Method to determine a plane using one point on the plane and two independent vectors parallel to it.

Point-Normal Form of a Plane

r - ro • n = 0, where ro is a point on the plane and n is a normal vector.

Cross Product Definition

The cross product of vectors x and y in 3-space produces a vector orthogonal to both x and y.

Area of a Parallelogram by Cross Product

The area of the parallelogram formed by vectors x and y is given by ||x × y||.

Scalar Triple Product

The volume of a parallelepiped determined by three vectors x, y, and z is given by |x • (y × z)|.

Intersection of Line and Plane

To find where a line intersects a plane, substitute the line's parametric equations into the plane's equation.