6.1 Vectors in the Plane

6.1 Vectors in the Plane

Some quantities can be represented by numbers only: Temperature, height, distance, volume, speed, area. These numbers indicate the size or magnitude.

Other quantities have both size (magnitude) and direction: velocity, acceleration, force.

VectorWe represent magnitude and direction with an arrow (directed line segment) whose tail is at the origin and an ordered pair determines the head. (𝑎 ,𝑏 )

This is the position vector of and is denoted by or

is known as the component form of vector

Vectors do not have to originate from the origin. As long as two vectors have the same length and same direction they are equivalent.

and have the same length and direction even though it is in a different location (position). and are equivalent and

A

B

C

D

𝑣

�⃗�

⟨5 ,2 ⟩

(−1 ,3 )

(4 ,5 ) Terminal – initialOr

Head – tail

Given the points and B, the vector with representation is

Ex: Find the vector represented by the directed line segment with initial point and terminal point

MagnitudeIf is represented by the arrow from to then

If then

ExamplesFind the magnitude of where and

Find the component form of

The Sum of Two VectorsIf a particle moves from A to B then changes direction and moves to C, the resulting displacement is from A to C.

A

B

C is the resultant

Definition of Vector AdditionIf and are vectors positioned so the initial point of is at the terminal point of (tip to tail) then the sum of is the vector from the initial point of to the terminal point of . The resultant (sum) is from the tail of the first to the head of the last.

Definition of scalar multiplicationIf c is a scalar (magnitude only) and is a vector, then the scalar multiplication is the vector whose length is times the length of and whose direction is the same as if and is opposite to if .

Math with Components

Examples

Find Find the magnitude of

Vector Operations

Find

Find

How would you draw ?

Given vectors and

Show:

�⃗� �⃗�

ExampleIf and

Find:

Unit VectorsA unit vector is a vector of length 1 in the direction of the original vector

Find the unit vector of

Day 2 – Vectors

Component form to linear combination

-3 is the horizontal component2 is the vertical component

-3

2

How would we find magnitude and direction?

𝑣

Components

𝑣

xθ component

y component

𝑣=⟨𝑥 , 𝑦 ⟩

ExampleFind the components of with direction angle of and magnitude of 6

Find the magnitude and direction of

Velocity vs SpeedVelocity is a vector. It has magnitude and direction. Speed is the magnitude of the velocity – no direction! It is not a vector. It is a scalar only.

ExampleA DC – 10 jet aircraft is flying on

a bearing of at 500 mph. Find the component form of the velocity of the airplane.

Example

Ex: A 100 lb weight hangs from two wires. Find the tensions T1 and T2 in both wires and their magnitudes.

50° 32°

100 lb

T1 T2