Two - Dimensional (2D) (Notes)

Two-dimensional (2D) space is one of the fundamental concepts in geometry, algebra, calculus, and other branches of mathematics. It refers to a flat plane in which any point can be described using two coordinates. Below is a detailed explanation of 2D space in mathematics, including its features, key concepts, and examples.

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1. Definition of 2D Space

In mathematics, 2D space refers to a plane that is defined by two dimensions: length and width (or height). This is usually described by a pair of coordinates $(x,y)$

 where:

• x-coordinate represents the horizontal position.
• y-coordinate represents the vertical position.

A point in 2D space is written as an ordered pair $(x, y)$in the Cartesian coordinate system or as $(r, \theta)$ in the polar coordinate system.

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2. Coordinate Systems in 2D

There are different ways to represent points in 2D space. The two most common systems are Cartesian coordinates and polar coordinates.

a) Cartesian Coordinate System

• Points are defined using two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical).
• Each point in 2D space is written as $(x, y)$ where:
  • x is the distance along the x-axis.
  • y is the distance along the y-axis.

b) Polar Coordinate System

• Points are represented by their distance from a reference point (called the origin) and an angle from a reference direction (typically the positive x-axis).
• Each point is described by $(r, \theta)$ where:
  • r is the radial distance from the origin.
  • θ is the angle measured counterclockwise from the positive x-axis.

Conversion between Cartesian and Polar Coordinates:

$$x = r \cos \theta \quad , \quad y = r \sin \theta$$
$$r=\sqrt{x^2+y^2},\theta ={\tan }^{-1}\left(\frac{y}{x}\right)$$

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3. Geometric Shapes in 2D

Various shapes exist in 2D space, each with specific properties related to angles, sides, and symmetry.

a) Lines and Line Segments

• A line is an infinitely extending straight path in both directions with no endpoints.
• A line segment is a portion of a line with two fixed endpoints.

The equation of a line in the Cartesian plane is usually written in slope-intercept form:

$$y=mx+b$$

where:

• $m$ is the slope of the line.
• $b$ is the y-intercept (the value of y where the line crosses the y-axis).

b) Basic 2D Shapes

• Circle: Set of points in a plane that are equidistant from a central point.
  • Equation: $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center, and $r$ is the radius.
• Triangle: A polygon with three sides and three angles.
  • Types include equilateral (all sides equal), isosceles (two sides equal), and scalene (no equal sides).
• Rectangle: A quadrilateral with four right angles. Opposite sides are equal in length.
• Square: A special type of rectangle where all sides are equal.
• Polygon: A closed shape with straight sides, such as pentagons, hexagons, and octagons.

c) Perimeter and Area

The perimeter is the total length of the boundary of a 2D shape, while the area is the amount of space enclosed within the shape.

• Perimeter of a Rectangle: $P = 2(l + w)$
  
• Area of a Rectangle: $A = l \times w$
  
• Perimeter of a Circle (Circumference): $C = 2 \pi r$
  
• Area of a Circle: $A = \pi r^2$
  

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4. Transformations in 2D

Transformations are operations that move or change shapes in the plane without altering their intrinsic properties (such as shape and size).

a) Translation

• Moves a shape from one position to another without rotating or resizing it. The shape is shifted by adding a constant vector $(a, b)$to every point.

$$(x^{\mathrm{\prime }},y^{\mathrm{\prime }})=(x+a,y+b)$$

b) Rotation

• Rotates a shape around a fixed point (usually the origin) by an angle $\theta$.
  • The new coordinates after rotation are given by:
  $$x' = x \cos \theta - y \sin \theta$$
  $$y' = x \sin \theta + y \cos \theta$$
  

c) Reflection

• Reflects a shape across a line (e.g., the x-axis or y-axis).

d) Scaling

• Enlarges or reduces the size of a shape by multiplying each coordinate by a constant factor $s$.

$$(x', y') = (sx, sy)$$

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5. Vectors in 2D

A vector in 2D is an object with both magnitude and direction. Vectors are usually written as:

$$\vec{v} = (v_x, v_y)$$where:

• $v_x$and $v_y$ are the components of the vector along the x-axis and y-axis, respectively.

Vector Operations:

• Addition: $\vec{v}+\vec{w}=(v_x+w_x,v_y+w_y)$
• Scalar Multiplication: $k \vec{v} = (k v_x, k v_y)$
  
• Dot Product: $\vec{v} \cdot \vec{w} = v_x w_x + v_y w_y$ which gives a scalar value.

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6. Calculus in 2D

In multivariable calculus, we often deal with functions of two variables, $z = f(x, y)$. These functions describe surfaces in 3D, but in a 2D context, they represent contour plots or cross-sections of surfaces.

Partial Derivatives:

Given $z = f(x, y)$, the partial derivatives $\frac{\partial f}{\partial x}$ and $\frac{\partial f}{\partial y}$ represent the rates of change of $z$ with respect to $x$ and $y$.

Gradient:

The gradient of a scalar function $f(x, y)$ is a vector that points in the direction of the steepest increase of the function. It is defined as:

$$\nabla f = \left( \frac{\partial f}{\partial x}, \frac{\partial f}{\partial y} \right)$$

Line Integral:

In 2D, a line integral computes the integration of a function along a curve $C$. For a vector field $\vec{F}(x, y)$ the line integral is:

$$\int_C \vec{F} \cdot d\vec{r} = \int_a^b \left( P(x, y) \frac{dx}{dt} + Q(x, y) \frac{dy}{dt} \right) dt$$

where $P(x, y)$ and $Q(x, y)$ are the components of $\vec{F}$ and $\vec{r}(t)$ parameterizes the curve.

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7. Trigonometry in 2D

Trigonometry deals with the relationships between angles and sides of triangles in 2D space. The basic trigonometric functions are:

• Sine: $\sin \theta = \frac{opposite}{hypotenuse}$
• Cosine: $\cos \theta = \frac{adjacent}{hypotenuse}$
• Tangent: $\tan \theta = \frac{opposite}{adjacent}$

These functions are essential for solving problems involving angles, distances, and periodic phenomena in 2D space.

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Applications of 2D in Mathematics

• Geometry: Study of shapes, areas, and distances on a plane.
• Graph Theory: Representing networks or connections using vertices and edges in 2D.
• Vector Calculus: Calculations with fields and functions defined in 2D.
• Linear Algebra: Matrices and transformations in two dimensions.
• Physics: Describing motion and forces restricted to a plane, such as projectile motion.

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Here are 5 examples of two-dimensional (2D) space in mathematics:

1. Equation of a Line:

• Example: The equation of a line in a 2D Cartesian plane is given by the linear equation: $$y = 2x + 3$$
  
  • This represents a straight line where the slope is 2, and the y-intercept is 3. Any point (x, y) that satisfies this equation lies on the line.

2. Distance Between Two Points:

• Example: Given two points $A(1, 2)$ and $B(4, 6)$ in a 2D plane, the distance between them is calculated using the distance formula: $$d=\sqrt{(x_2-x_1{)}^2+(y_2-y_1{)}^2}=\sqrt{(4-1{)}^2+(6-2{)}^2}=\sqrt{9+16}=\sqrt{25}=5$$
  • The distance between points A and B is 5 units.

3. Area of a Triangle:

• Example: For a triangle with vertices at $A(1, 1)$, $B(4, 1)$, and $C(4, 5)$, the area can be calculated using the formula: $$\text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right|$$ Plugging in the coordinates of A, B, and C: $$\text{Area} = \frac{1}{2} \left| 1(1 - 5) + 4(5 - 1) + 4(1 - 1) \right| = \frac{1}{2} \left| -4 + 16 + 0 \right| = \frac{1}{2} \times 12 = 6 \, \text{square units}$$
  
  • The area of the triangle is 6 square units.

4. Circle Equation:

• Example: The equation of a circle with a radius of 5 units centred at the point $(2, 3)$is given by: $$(x - 2)^2 + (y - 3)^2 = 25$$
  
  • Any point $(x, y)$ that satisfies this equation lies on the circle with a radius of 5, centered at $(2, 3)$.

5. Parabola:

• Example: The equation of a parabola is given by $y = x^2 - 4x + 3$, which is a quadratic function.
  • To find the vertex of the parabola, we complete the square or use the vertex formula:
  $$x = -\frac{b}{2a} = -\frac{-4}{2(1)} = 2$$Substituting $x=\mathrm{2\; into\; the\; equation:}$$$y = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1$$
  
• The vertex of the parabola is at $(2, -1)$, and the curve opens upwards.

Conclusion

Two-dimensional space in mathematics is a crucial concept that forms the foundation for understanding geometry, algebra, trigonometry, calculus, and many other fields. From simple shapes to complex transformations, 2D space provides a framework for analyzing and solving a wide variety of mathematical problems.