Multivariable Calculus Concepts: Planes, Spheres, and Vectors

Vocabulary flashcards reviewing 3D planes, standard equations and geometric properties of spheres, vector magnitudes, and dot products based on the lecture transcript.

Plane parallel to the $yz$-plane

A plane passing through a point $(a, b, c)$ that is parallel to the $yz$-plane, defined by the equation $x = a$.

Plane parallel to the $xz$-plane

A plane passing through a point $(a, b, c)$ that is parallel to the $xz$-plane, defined by the equation $y = b$.

Plane parallel to the $xy$-plane

A plane passing through a point $(a, b, c)$ that is parallel to the $xy$-plane, defined by the equation $z = c$.

Standard equation of a sphere

The algebraic representation $(x - a)^2 + (y - b)^2 + (z - c)^2 = r^2$, where $(a, b, c)$ is the center of the sphere and $r$ is the radius.

Center of a sphere from diameter endpoints

The midpoint of the line segment connecting two endpoints of a diameter, found by adding corresponding components of each point and dividing by $2$.

Magnitude of a vector

The length of a vector, which represents the distance between two points in space.

Radius of a sphere from diameter endpoints

Half of the magnitude of the vector formed between the two endpoints of the diameter of the sphere.

Dot product

An operation on two vectors that multiplies corresponding components together and sums the products, resulting in a single scalar number.

Scalar product

An equivalent term for the dot product of two vectors, named because its result is a scalar value rather than a vector.