12.2 Vectors

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

 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

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

and If then v is a zero vector : are called the standard basis 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. 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. Parallelogram Law

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E Application

N E u

N E v u 60 o

A Boeing 727 airplane, flying due east at 500mph in still air, encounters a 70-mph tail wind acting in the direction of 60 o north of east. The airplane holds its compass heading due east but, because of the wind, acquires a new ground speed and direction. What are they? N E v u We need to find the magnitude and direction of the resultant vector u + v. u+v

N E v u The component forms of u and v are: u+v Therefore: and: The new ground speed of the airplane is about mph, and its new direction is about 6.5 o north of east.

i j v is called a linear combination of i and j Any vectors can be written uniquely in terms of standard basis vectors : v (measured counterclockwise) with the positive x -axis then v can be written as If v is any nonzero vector that makes an angle Linear Combination

v is called a linear combination of i, j and k Standard basis vector notation Linear Combination in Space

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