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vector

adding vectors

subtracting vectors

scalar multiplication

quantity with size and direction, run and rise

add numbers in corresponding components

subtract numbers in corresponding components

multiplying a vector times a number

vector

quantity with size and direction, run and rise

adding vectors

add numbers in corresponding components

Vector

speed

Scalar

Position Vector

quantity characterized by magnitude and direction

the rate of change of position with respect to time

A constant value, a quantity that only has magnitude

A vector that starts from origin in space

Vector

quantity characterized by magnitude and direction

speed

the rate of change of position with respect to time

What is a vector quantity?

What is displacement?

What do the components of a vector des…

How can you find the resultant of a ve…

Magnitude and direction

Shortest distance from one point to another

Journey from start to end

Pythagoras theorem

What is a vector quantity?

Magnitude and direction

What is displacement?

Shortest distance from one point to another

i

j

(q₁-p₁, q₂-p₂)

√(q₁-p₁)²+(q₂-p₂)²

Represents the horizontal value of a vector in an equation.

Represents the vertical value of a vector in an equation.

Component Form of a Vector with initial Point P = (p₁, p₂) an…

Magnitude of a Vector with initial Point P = (p₁, p₂) and ter…

i

Represents the horizontal value of a vector in an equation.

j

Represents the vertical value of a vector in an equation.

Pythagoras' Theorem

Bearings

Trigonometry Triangles

Sine Rule

a^2 + b^2 = c^2... c = hypotenuse (longest side)... 3-D Pythagoras…

Always calculate angle clockwise from north and bearings are…

Use for triangles that are right angled. Remember by - SOH, C…

For non right angled triangles.... To find a side (given 2 angle…

Pythagoras' Theorem

a^2 + b^2 = c^2... c = hypotenuse (longest side)... 3-D Pythagoras…

Bearings

Always calculate angle clockwise from north and bearings are…

HYPOTENUSE

PYTHAGOREAN THEOREM (KNOW FORMULA)

NUMERATOR

DENOMINATOR

the longest side of a right triangle, opposite the right angle

In a right angled triangle the square of the long side is equ…

the number above the line in a common fraction

the number below the line in a common fraction

HYPOTENUSE

the longest side of a right triangle, opposite the right angle

PYTHAGOREAN THEOREM (KNOW FORMULA)

In a right angled triangle the square of the long side is equ…

what does this mean ||?

how do you find the magnitude of one v…

what does the position vector of B rel…

what does "find the position vector of…

find the magnitude

put it under the square root and square all the individual nu…

vector AB

NM

what does this mean ||?

find the magnitude

how do you find the magnitude of one v…

put it under the square root and square all the individual nu…

To describe motion in 2 or 3 dimension…

vector quantity

scalar quantity

sin ø = cos ¢ =

vectors

has both a magnitude (how much) and a direction

does not involve a direction

opposite÷hypotenuse

To describe motion in 2 or 3 dimension…

vectors

vector quantity

has both a magnitude (how much) and a direction

Vector

Scalar quantity

Magnitude of a vector

equal

a quantity that has both magnitude and direction

has magnitude only ex: speed, distance, temperature

the measure of the length (positive real number)

two vectors are ____ if their magnitudes and directions are t…

Vector

a quantity that has both magnitude and direction

Scalar quantity

has magnitude only ex: speed, distance, temperature

Relationships between vectors... - a dire…

Scalar multiplication

Vectors in rect. Coordinates

Unit vector

It has direction and magnitude... Magnitude- how long ... Direction…

If k is a real number and v is a vector kv is called a scalar…

Ai(x axis)+bj (y axis)... If initial point is 0,0 then it's call…

The vector v/magnitude of v is the unit vector that has the s…

Relationships between vectors... - a dire…

It has direction and magnitude... Magnitude- how long ... Direction…

Scalar multiplication

If k is a real number and v is a vector kv is called a scalar…

Terminal

Orthogonal

Unit Vector

Component Form

Point at the end of a vector

Two vectors whose dot product equals zero

Have a length of one

Form of a vector that is uniquely represented by the coordina…

Terminal

Point at the end of a vector

Orthogonal

Two vectors whose dot product equals zero

Terminal

Orthogonal

Unit Vector

Component Form

Point at the end of a vector

Two vectors whose dot product equals zero

Have a length of one

A vector that is uniquely represented by the coordinates of i…

Terminal

Point at the end of a vector

Orthogonal

Two vectors whose dot product equals zero

Vector

speed

Scalar

Position Vector

quantity characterized by magnitude and direction

the rate of change of position with respect to time

A constant value, a quantity that only has magnitude

A vector that starts from origin in space

Vector

quantity characterized by magnitude and direction

speed

the rate of change of position with respect to time

Vector

Terminal Point

Vertical Component

Initial Point

A quantity that has both magnitude and direction.

The ending point of a vector

The vertical change from the initial point to the terminal po…

Starting point of a vector

Vector

A quantity that has both magnitude and direction.

Terminal Point

The ending point of a vector

Component form of a vector is

To find component form, you need to re…

To get 2 vectors into one, you...

To find the magnitude you....

(V1, V2)

Terminal - Initial

add them together

√(〖V1〗^2+〖V2〗^2 )

Component form of a vector is

(V1, V2)

To find component form, you need to re…

Terminal - Initial

To find the component form of a vector…

To find the magnitude

To find the direction angle

Orthogonal means

Terminal-Initial

Take the square root of <x^2+y^2>

SET CALC TO DEGREES... Take the inverse tangent of (y/x)

Two Perpendicular Vectors... Occurs when the dot product of two…

To find the component form of a vector…

Terminal-Initial

To find the magnitude

Take the square root of <x^2+y^2>

conservative vector field definition

line integral over a scalar

line integral over a vector

fundamental theorem of calculus for li…

A vector field F is called conservative if it is the gradient…

∫f(x,y)ds over c =∫f(x(t),y(t)) √((dx/dt)²+(dy/dt)²)dt over a…

∫vec(F)•dr over c = ∫F(r(t))•r'(t)dt over a<t<b.

Let c be a smooth curve given by the vector function r(t), a<…

conservative vector field definition

A vector field F is called conservative if it is the gradient…

line integral over a scalar

∫f(x,y)ds over c =∫f(x(t),y(t)) √((dx/dt)²+(dy/dt)²)dt over a…

vector:

Initial point:

Terminal point:

the set of all directed line segments…

can used to represent a quantity that involves both magnitude…

point you start at

point you end at

vector

vector:

can used to represent a quantity that involves both magnitude…

Initial point:

point you start at

vector

magnitude

direction

component form

an ordered pair of real numbers that have direction and length

the length of the arrow v=√(a)²+(b)² or v=√(x₂-x₁)²+(y₂-y₁)²

the direction in which the arrow is pointing

<a,b> to find: <x₂-x₁,y₂-x₁>

vector

an ordered pair of real numbers that have direction and length

magnitude

the length of the arrow v=√(a)²+(b)² or v=√(x₂-x₁)²+(y₂-y₁)²

Derivative of vector valued function

Limits and vector valued functions

Vector valued functions, derivative ru…

Proofs of derivative rules?

r'(t)=(x'(t),y'(t), z'(t))

lim h->0 (r(t+h)-r(t))/h <--if this exists, then r'(h) exists

sum rule: derivative of each function, add ; constant rule: p…

split r(t) into components (x(t), y(t) ,z(t)) distribute scal…

Derivative of vector valued function

r'(t)=(x'(t),y'(t), z'(t))

Limits and vector valued functions

lim h->0 (r(t+h)-r(t))/h <--if this exists, then r'(h) exists

Magnitude of a vector

Addition of vectors

Ai+bj

How to make it a unit vector

Square root of v1 squared plus v2 squared

Add first ones, then add second ones

Can be written in component form as <I,j>

Divide by the magnitude of the vector

Magnitude of a vector

Square root of v1 squared plus v2 squared

Addition of vectors

Add first ones, then add second ones

resultant

opposite vectors

parallel vector

component

new vector after combining other vectors

sam magnitude and opposite directions

if same direction or opposite direction and if θ= 0 or π

two or more vector whose sum is a given vector

resultant

new vector after combining other vectors

opposite vectors

sam magnitude and opposite directions

Formula for Length(Magnitude) of a Vec…

Formula for Distance between two vecto…

Formula for Normalizing a vector?

Common use of Dot Product in Games?

sqrt(x^2 + y^2 + z^2)

First -Subtract Vectors (x1,y2,z2)-(x2,y2,z2) = NV (NewVector…

divide each component(x,y,z) by the vector's length.

Finding if an object is in front of or behind another object.…

Formula for Length(Magnitude) of a Vec…

sqrt(x^2 + y^2 + z^2)

Formula for Distance between two vecto…

First -Subtract Vectors (x1,y2,z2)-(x2,y2,z2) = NV (NewVector…

Position Vector

Parallel Vectors

Magnitude

Unit Vector

P = (a, b, c) O->P = (a)... (b)... (c)

Same direction but magnitude scalar multiples of each other

P = (a) ｜P｜= (a^2 + b^2 + c^2)^1/2... (b)... (c)

Vector with magnitude 1

Position Vector

P = (a, b, c) O->P = (a)... (b)... (c)

Parallel Vectors

Same direction but magnitude scalar multiples of each other

Additive Closure (AC)

Scalar Closure (SC)

Commutativity (C)

Additive Associativity (AA)

if u, v ∈ V, then u + v ∈ V

If α ∈ ℂ and u ∈ V, then αu ∈ V.

if u, v ∈ V, then u + v = v + u

if u, v, w ∈ V, then u + (v + w) = (u + v) + w

Additive Closure (AC)

if u, v ∈ V, then u + v ∈ V

Scalar Closure (SC)

If α ∈ ℂ and u ∈ V, then αu ∈ V.

Area of a triangle normal

Trigonometry area of a triangle

The sine rule for working out missing…

Sine rule for working out missing angles

(Base x height)➗2

1/2absinC

a over sineA = b over sineB = c over sineC

SinA over a = SinB over b = SinC over c

Area of a triangle normal

(Base x height)➗2

Trigonometry area of a triangle

1/2absinC

vector

initial point

terminal point

equivalent vectors

quantity with specific magnitude and direction (these require…

starting point of a vector

end point of a vector (has the arrowhead)

same magnitude and direction

vector

quantity with specific magnitude and direction (these require…

initial point

starting point of a vector

Vectors have what?

Magnitude of a vector?

Scalar multiplication of a vector

Represent vectors using rectangular co…

Magnitude and direction

Is it's length. Formula v = √(a^2+b^2)

Multiplying a vector by a scalar changes the magnitude of the…

V=(x2-x1)I+(y2-y1)j

Vectors have what?

Magnitude and direction

Magnitude of a vector?

Is it's length. Formula v = √(a^2+b^2)

Vectors

Scalars

Vector addition

Component vectors equation

are physical quantities that have both magnitude and directio…

are quantities without direction. Scalar quantities may be th…

may be accomplished using the tip-to-tail method or by breaki…

SOH CAH TOA.... In other words: X = V cosθ... Y = V sinθ... Where V i…

Vectors

are physical quantities that have both magnitude and directio…

Scalars

are quantities without direction. Scalar quantities may be th…