Understanding Vector Space
Build a cartesian system from first principals
Table des matières
- Euclid's Elements - Definitions
- Definitions
- Create Points
- Create a Circle
- Euclidian Tools
- Create a Parallel straight lines
- Vectors
- Vector Tools
- Euclidian Tools
- Scale a Vector
- Create a Vector Space
- Create a cartesian system
- Create a cartesian system
- Review
- Congruent Figures: Dynamic Illustration
- Congruent Circles: Definition
- Kala - Exploring Congruent Triangles SSS ASA SAS SSA
- Congruent Triangles SAS
- SSS
- Demo ASA
Definitions
Definition 1. A point is that which has no part.
A point is a location in the plane with no dimensions, represented as a dot and denoted by a capital letter A, B, P, ...[br]If two points are proven to be in the same location, they are the same point.
Figure 1: Example of points
Figure 1: Examples of Points
[b]A[/b], [b]B[/b], and [b]C[/b] are points in the plane.[br][math]A_1[/math] and [math]B_1[/math] are connected by the vector [math]A_1B_1[/math][br][b]O[/b] is the centre of the circle with radius [b]OP[/b][br]
Definition 2: A Line
[b][url=https://mathcs.clarku.edu/~djoyce/java/elements/bookI/defI2.html]Definition[/url] [/b]A [i]line[/i] is breadthless length.[br][b][url=https://mathcs.clarku.edu/~djoyce/java/elements/bookI/defI3.html]Definition[/url] [/b]The ends of a line are points.[br][b][url=https://mathcs.clarku.edu/~djoyce/java/elements/bookI/defI4.html]Definition[/url] [/b]A [i]straight line[/i] is a line which lies evenly with the points on itself.[br][br]We will refer to a ``line'' in this work, we mean a straight-line[br]A line is a straight, one-dimensional object that extends infinitely in both directions. It has no thickness and is determined by any two distinct points on it. A line is denoted by two capital letters sometimes shown with a line above them, e.g., AB, CD, OP, ...
Figure 2: Examples of Lines
Figure 2, Examples of Line
[b]AB [/b]is a line segment, a portion of a line with two endpoints.[br][b]CD [/b]is an infinite line, extending indefinitely in both directions.[br][b]EF [/b]is a vector, a directed line segment from point E to point F.[br][b]OP [/b]is a radius, a line segment from the center O to a point P on the circle.
Definition 3: The Radius is a Fixed Distance from the Centre of a Circle
[b][url=https://mathcs.clarku.edu/~djoyce/java/elements/bookI/defI15.html]Definition[/url] [/b]A [i]circle[/i] is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure equal one another.[br][b][url=https://mathcs.clarku.edu/~djoyce/java/elements/bookI/defI15.html]Definition[/url] [/b]And the point is called the [i]center[/i] of the circle.[br][b][url=https://mathcs.clarku.edu/~djoyce/java/elements/bookI/defI15.html]Definition[/url] [/b]A [i]diameter[/i] of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.[br][br]A circle is a circumference, all points of the circumference are equal radii from the center[br]The distance from the center to the circle is called the radius.[br]A circle is denoted by the symbol ⊙ followed by its center, [br]e.g., ⊙AB, or ⊙(O,P) where OP is the radius of the circle and O is the center of the circle
Figure 3: Examples of Circles
Figure 3: Examples of Circles
⊙AB, is a floating circle where AB is the radius of the circle and A is the center of the circle [br]⊙(O,P) is defined by the line OP. Where OP is the radius and O is the center of the circle[br]⊙(C,D) is defined by Vector CD, where CD is the radius and C is the center of the circle
Perpendicular Lines
[b][url=https://mathcs.clarku.edu/~djoyce/java/elements/bookI/defI10.html]Definition[/url] [/b]When a straight line standing on a straight line makes the adjacent angles equal to one another, each of the equal angles is [i]right,[/i] and the straight line standing on the other is called a [i]perpendicular[/i] to that on which it stands.[br][br]Perpendicular lines are two lines that intersect at a right angle (90 degrees). Perpendicular lines are denoted by the symbol ⊥, e.g., AB⊥CD.
Figure 4: Example of Perpendicular Lines
Figure 4: Perpendicular Lines
Line AB is perpendicular to line CD[br]The space is split into 4 angles of equal size. [br]∠ AED = ∠ DEB = ∠ BEC = ∠ AEC [br]All Angles are 90 degrees
Definition 5: Parallel Line
[b][url=https://mathcs.clarku.edu/~djoyce/java/elements/bookI/defI23.html]Definition[/url] [/b][i]Parallel[/i] straight lines are straight lines which, being in the same plane and being produced indefinitely in both directions, do not meet one another in either direction.[br][br]Parallel lines are two or more lines in the same plane that never intersect, no matter how far they are extended in either direction. Parallel lines are denoted by the symbol ∥, e.g., AB∥CD
Vector Tools
Figure 3.1- Scalar increase
Figure 3.1: Vector AB is given.
[br]Circle C (center C) is not touching AB.[br]Using a compass and parallel line, vector CD is constructed equal to vector AB.
Figure 3.2 - Vector Sum
Figure 3.2: Vector AB and vector CD are given.
[br]Circle E (center E) is used to construct vector EF equal to vector AB (same length and same direction).[br]Circle F (center F) is used to construct vector FG equal to vector CD (same length and same direction).
Question 3.2
Given that EF = AB and FG = CD, what does EG represent?
Figure 3.3 - Scale vector by 1/n
[br]Extend line g from point B[br]Draw a circle around B with point D on g[br]Create the required n scalar value e.g. n =6 then 1/6AB requires 6DB[br][br]Drop a line from P_0 to A[br][br]Then Drop a parallel line from P_1, P_2, P_3, …, P_n to AB[br][br]This will cut AB into sections of 1/n[br][br][br]
Question 3.3.1
Using a compass, describe how you would construct point P₂ on line BP₁ given point P₁.
Question 3.3.2
Explain why the parallel lines from P₁, P₂, P₃, …, Pₙ divide AB into n equal sections.
Congruent Figures: Dynamic Illustration
[color=#0000ff]Recall an ISOMETRY is a transformation that preserves distance.[/color] So far, we have already explored the following isometries:[br][br][color=#0000ff]Translation by Vector[br]Rotation about a Point[br]Reflection about a Line[br]Reflection about a Point (same as 180-degree rotation about a point) [/color][br][br]For a quick refresher about [color=#0000ff]isometries[/color], see this [url=https://www.geogebra.org/m/KFtdRvyv]Messing with Mona applet[/url].
CONGRUENT FIGURES
[b]Definition: [br][br]Any two figures are said to be CONGRUENT if and only if one can be mapped perfectly onto the other using [color=#0000ff]any 1 or composition of 2 (or more) ISOMETRIES.[/color][/b][br][br]The applet below dynamically illustrates, [b]by DEFINITION[/b], what it means for any 2 figures (in this case, triangles) to be [b]CONGRUENT.[/b] [br][br]Feel free to move the BIG WHITE VERTICES of either triangle anywhere you'd like at any time.