PreCalc 2/8/21

Direction angles

- The direction of a vector is the angle that it makes with 0 degrees, the horizontal axis (unless you are working with bearings) - All angles in a problem must originate from the same place, either from the horizontal axis or from North To solve: tanθ=sinθ/cosθ = y/x θ = tan^-1 (y/x) - when graphing arrow goes counterclockwise from the x axis - add 180 degrees if vector is in Q3 or 4 - add 360 degrees if vector is in Q2

Graphs/Application word problems

1) Write each vector in component form 2) Perform vector addition to find the resultant 3) Identify the resultant vector as well as the magnitude and direction angle of the resultant Ex: v1 -> θ1 = x° IIv1II = y (magnitude of vector 1) Then find vector in component form -> v1=<ycosx°, ysinx°> Then add the vectors together in component form = Resultant vector Find the magnitude of the resultant vector R=<a,b> IIRII=sqrt(a)^2+(b)^2 = c (with unit; m/s or K, N) Find the magnitude and direction angle (vector angle) θ=tan^-1(b/a) - Based off of resultant vector if the first value is positive and the second is negative than add 360° to the magnitude and direction angle - Based off of resultant vector is both values are negative or the first value is negative and the second is positive than add 180° to the magnitude and direction angle

Vocab

A DIRECTED LINE SEGMENT can be used to represent a quantity that involves both magnitude and direction The directed line segment PQ has INITIAL point P and TERMINAL point Q The MAGNITUDE of the directed line segment PQ is denoted by IIPQII The set of all directed line segments that are equivalent to a given directed line segment PQ is a VECTOR v in the plane The directed line segment whose initial point is the origin is said to be in STANDARD POSITION The 2 basic vector operations are scalar MULTIPLICATION and vector ADDITION The vector u+v is called the RESULTANT of vector addition The vector sum v1i+v2j is called a LINEAR COMBINATION of vectors i and j and the scalars v1 and v2 are called the HORIZONTAL and VERTICAL components of v1 respectively Two directed line segments that have the same magnitude and direction are equivalent A unit vector has a magnitude of 1

Scalar quantities

Area, time, and temp are represented by a single real number

Bearings and applications

Bearings: Every angle must originate from the same direction to find the resultant N = 0° E = 90° S = 180° W = 270° Ways to describe the direction: N 40° E (from North go East 40°) NE 40° (from North go East 40°) 40° south of east (130° from North) -> 180-40 40° north of east (50° from north) -> 90-40 Due ___ = directly in that direction 30° northwest = from north go 30° west

Component to linear

Component: <9, -2> Linear: 9i - 2j

Vector quantities

Force and velocity involve both magnitude (distance/length) and direction (angle/bearing) which can't be represented by a single real number. Vectors are represented by a directed line segment (geometry figure). (Initial point {P} to terminal point{Q}) - IIPQII denotes the magnitude or length which can be determined using the distance formula - the set of all directed line segments equivalent to a given directed line segment PQ is a vector v in the plane - vectors are denoted by lowercase, bold letters (ex: u, v, w) - if two vectors have the same magnitude (length) and direction (angle) than u=v (equivalent)

Finding vector when given the direction angle and magnitude:

IIuII<cosθ,sinθ> or <IIuIIcosθ, IIuIIsinθ> can be simplified to <x,y> using vector operations Ex: Given the magnitude of the vector is 17 with a direction angle of 145 degrees, write the vector (find it in component form) <17cos145, 17sin145> = <-13.93, 9.75>

Finding the magnitude of a vector (length of v)

IIvII= sqrt(q1-p1)^2 + (q2-p2)^2 = sqrt(v1)^2+(v2)^2 sqrt(v1)^2+(v2)^2 -> v1 and v2 would be the component form of the vector sqrt(q1-p1)^2 + (q2-p2)^2 -> when component form has not been found Ex: Component form = <3,2> Magnitude = sqrt3^2+2^2 = sqrt13 ~ 3.61

Linear to component

Linear: 5i+6j Component: <5,6>

Component form of Vector/Finding the component form

The component form of vector with initial point P = (p1, p2) and terminal point Q = (q1, q2) is given by: PQ = <q1-p1, q2-p2> = <v1, v2> = v Ex: Identify the component form of a vector that has an initial point of (2,7) and a terminal point of (-4,6) P: (2,7) Q: (-4,6) Component: <-4 -2, 6-7> = <-6,-1> Ex: u is a directed line segment from P = (0,0) to Q = (3,2) Find the component form of vector u: <3-0, 2-0> u= <3,2>

Resultant Vector (Answer vector)

The resultant vector is vector that "results" from adding two or more vectors together. The resultant vector goes from the initial point of the first vector to the terminal point of the last vector

Represent the following vector operations graphically:

Will be given two vectors; graph them using first vector addition or scalar multiplication then use the tail to tip method Ex: Let v=<-2,5> and w=<3,4>, find the following vectors 1) 2v 2)w-v Solve and then graph

Vector addition and scalar (distributive property) multiplication (Linear combination form)

i=<1,0> and j=<0,1> are standard unit vectors v=<v1, v2> vector v written in component form v= v1i+v2j vector v written in linear combination Ex: Let v= <-2, 5> and w= <3,4> find the following vectors: v-2w <-2,5>-2<3,4> <-2,5>+<-6,-8> <-8,-3>