1.1 – 1.2 The Geometry and Algebra of Vectors

 Quantities that have magnitude but not direction are called scalars. Ex: Area, volume, temperature, time, etc.  Quantities such as force, acceleration or velocity that have direction as well as magnitude are represented by directed line segments, called vectors. A B Initial point (tail) terminal point (head)  The length of the vector is called the magnitude and is denoted by Vectors in the Plane

 A vector is in standard position if the initial point is at the origin. x y  The component form of this vector is:  Vectors are equivalent if they have the same length and direction (same slope). then the component form of is:  If are initial and terminal points of a vector, P Q (c,d) (a,b) v (a-c, b-d) x

P Q (-3,4) (-5,2) The component form of is: v (-2,-2) The magnitude is Example

e1e1 and e2e2 If then v is a zero vector: are called the standard unit vectors. The magnitude ofis: If then v is a unit vector.

Vector sum: Vector difference Scalar Multiplication: Negative (opposite): Vector Operations

v v u u u+v u + v is the resultant vector. v v u u u-v u - v is the resultant vector. The Parallelogram Rule (The Head-to-Tail Rule)

Linear Combination

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

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 in Space

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

The set of all vectors with n real-valued components is denoted by R n. Thus, a vector in R n has the form  R 2 is the set of all vectors in the plane.  R 3 is the set of all vectors in three-dimensional space.  Z 3 is the set of all vectors in three-dimensional space whose components are integers.  Z 3 is the set {0, 1, 2} with special operations (Integer modulo 3) Vectors in R n and Z n

Algebraic Properties of Vectors

The dot product of u and v is defined by Two vectors u and v are orthogonal  if they meet at a right angle.  if and only if u ∙ v = 0. (Read “u dot v”) The dot product is also called scalar product. Definitions The distance between vectors u and v is defined by

Example

Another form of the Dot Product: Properties

Find the angle between vectors u and v : Example

Let u and v be nonzero vectors.  w 1 is called the vector component of u along v (or projection of u onto v ), and is denoted by proj v u  w 2 is called the vector component of u orthogonal to v w2w2 w1w1 u v Vector Components

1.Write u as a linear combination of standard unit vectors. 2.Find u + v and 2 u – 3 v. 3.Are u and v parallel? orthogonal? 4.Find the angle between u and v. 5.Find the magnitude of v. 6.Normalize vector v. 7.Find the projection of u onto v. 8.Find the vector component of u orthogonal to v. 9.Find the projection of v onto u. 10.Find the distance between u and v. Examples