11.2 Vectors in Space

A three-dimensional coordinate system consists of:  3 axes: x-axis, y-axis and z-axis  3 coordinate planes: xy -plane, xz -plane and yz -plane  8 octants. Each point is represented by an ordered triple x z y Each vector is represented by Coordinates in Space

Midpoint Formula: Distance Formula: Given 2 points in the space with coordinates Standard equation of a sphere of radius r, centered at is

1) Find the standard equation of the sphere whose endpoints of a diameter are 2) Find the center and radius of the sphere: 3) Describe the solid satisfying the condition: Examples

and If then v is a zero vector : are called the standard unit vectors. The magnitude ofis: If then v is a unit vector. Vectors

Vector sum: Vector difference Scalar Multiplication: Negative (opposite):  Vector v is parallel to u if and only if v = ku for some k. Vector Operations

1) Find the unit vector in the direction of v 2) Determine whether the points are collinear: 3) Show that the following points form the vertices of a parallelogram: Examples (Normalize v )

 v is called a linear combination of i, j and k Standard unit vector notation  Unit vector in the direction of v is given by This unit vector is called the normalized form of v. Linear Combination