Operations with Vectors
What is a vector?
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
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A quantity that has both
1. Size
2. Direction
Examples
1. Wind
2. Boat or Aircraft travel
3. Forces in Physics
Geometrically
A line segment
In a more mathematical aspect…
A vector is a quantity has direction as well as magnitude, especially as determining the position
of one point in space relative to another.
It is shown as an arrow
the length = magnitude and the arrow head’s direction = direction
Vectors are scalar quantities, which means that they are real numbers and has magnitude but not
direction, and are commonly expressed as “v”
The magnitude (resultant) of two vectors is the hypotenuse of a triangle
Vector Notation
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Vectors can be given by
1. Angle brackets Ex. < a, b > with an initial and terminal point
2. Ordered Pair Ex. ( a, b ) with an initial and terminal point
Adding Vectors
Vectors can be added algebraically and graphically
See Textbook pages (599 – 602) for references
Algebraically
Given <1, 2> + <7, 0> = <8, 2>
X values are added to other x values & Y values are added to other y values
When adding vectors, all of the vectors must have the same units.
All of the vectors must be of the same type of quantity.
Graphically
Given <1,2>+<7,0>
Starting from the orgin draw your first vector 1 being your x value and 2 being your y value
Then from the TIP of your first vector , draw your second one ( tip to tail method )
Examples!!!!!!!!!
Let p=<1, 2> and q=<7, 0> Find each of the following
1. p + q
First substitute numbers for vectors <1, 2 > +<7, 0>
2. Combine like terms <1+7, 2+0>
3. Solve p + q = <8,2>
Subtracting Vectors
Vectors can also be subtracted algebraically or graphically
See textbook pages (599-602) for reference
Algebraically
Given <5,8>-<1,3>=<4,5>
Subtract corresponding components ex. 5-1=4 and 8-3=5
X values are added to other x values & Y values are added to other y values
When adding vectors, all of the vectors must have the same units.
All of the vectors must be of the same type of quantity.
Graphically
Given <5,8>-<1,3>
First you must change you operation to addition and change one of your vectors to
negative
Making your new problem <5,8>+<-1,-3>
Now you are ready to graph
Starting from the orgin draw your first vector 1 being your x value and 2 being your y
value
Then from the TIP of your first vector, draw your second one (tip to tail method)
Examples!!!!!!!!!
Let p=<5, 8> and q=<1, 3> Find each of the following
1. p - q
First substitute numbers for vectors <5, 8 > +<1, 3>
2. Combine like terms <5-1, 8-3>
3. Solve p - q = <4,5>
Multiplying Vectors
Vectors can only be multiplied by scalars meaning that the direction of the vector wont
change but the magnitude will
See text book pages (599-602)
Algebraically
When multiplying vectors use distributive property
Given 3*<3,4> = <9,12>
Graphically
Given 3*<3,4> = <9,12>
Multiplying vectors is as simple as adding the same vector from tip to tail as many times
as the original vector is being multiplied by.
For example, since <3,4> is being multiplied 3 times, you would add that same vector on
top of each other 3 times.
Example!!!!!!!
Let g=<3,4> and h=3 Find each of the following
1. g*H
First substitute numbers for vectors <3,4> 3
2. Distribute outside number into the following vetcor
3. Solve g*H= <9,12>
Dividing Vectors
When dividing vectors they need to be in a specific form.
Polar Form, rcisΘ, is the short hand version of magnitude and direction combined
together. When calculating polar form r would equal the magnitude and Θ would equal
the direction. Once your vectors are in polar form they have a short hand way with
dividing.
See textbook pages 599-602 for a reference
Algebraically
When dividing in polar form, divide the magnitudes or “r” and subtract the directions or
“Θ”
Given 6cis75° / 3cis25° = 2cis50°
Example!!!!!!
1. First find the magnitude and direction of a vector r=6 Θ=75°
2. Repeat for the second vector r=3 Θ=25°
3. Acquire the rcisΘ “Short hand polar form” 6cis75° 3cis25°
4. Align them for division 6cis75°/ 3cis25°
5. Divide magnitude(s) and subtract direction(s) 6/3=2 75-25=50
6. Solve 6cis75°/ 3cis25°= 2cis50°
Quiz
A= 3 B= <1,6> C=<2,4> D=<7,3> E=<0,8> F=4
1. b+c
2. c+d
3. c-b
4. d-b
5. a*b
6. f*c
Answers
1. <1,6> + <2,4>= <3,10>
2. <2,4>+<7,3>= <9,7>
3. <2,4>+<-1,-6>= <1,-2>
4. <7,3>- <1,6>=<6,3>
5. 3<1,6>= <3,18>
6. 4<2,4>= <8,16>
References and links
Khanacademy
Studyisland
Desmos
www.youtube.com/watch?v=lQww6qzqD7g