Physics Vectors

Definition:

The descriptions of quantities with both magnitude and direction are called vectors. Examples include velocity, force, displacement, weight, and acceleration.

Representation:

The vector is represented by a line with an arrow. The point O from which the arrow starts is called the tail or starting point.  A point where the arrow ends is called the vertex, main, or endpoint. A vector moved parallel to itself remains unchanged. If it is rotated by an angle other than 360, it will change.

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Three forms in which vectors can be represented:

Component Form:

  In component form, it is represented by its components along the coordinate axes. As mentioned earlier, in a 2D Cartesian coordinate system a vector V can be represented as V = (Vx, Vy), and in a 3D Cartesian coordinate system it can be represented as V = (Vx, Vy, vz). The component shape allows you to split it into its parts in different directions.

The magnitude and direction shape: In this shape, it is represented by its magnitude (or length) and direction. Instead of specifying the components, we describe them using the length and angle or direction from the base axis. For example, a vector V with a magnitude of 5 units and an angle of 30 degrees from the positive x-axis can be represented as V = 5 units, 30 degrees. This shape is especially useful when working with geometric interpretations or when working in polar or spherical coordinate systems.

Form of Unit Vector: A unit vec. is that whose magnitude is 1. In this form, it is represented by a unit vector in the direction of the original vec. This is often indicated by placing a hat symbol (^) over the symbol. For example, if V is a vector, the unit form is V̂. This form is useful when you need to focus on the direction rather than its magnitude.

Types:

There are many types, here is a comprehensive list of additional types:

Zero vector: VCTR with magnitude zero, also rep presented as  0.

Unit Vec: A vector of magnitude 1. It is often used to indicate direction.

Position Vec: This represents the position of a point relative to a control point or origin.

Common starting vector: two or more VCTRs that share the same starting point or origin.

Similar and Different Vec: They are those that have the same direction regardless of their size. Unequal vectors have the same magnitude but opposite directions.

Coplanar Vec: Three or more vectors lying in the same plane.

Collinear Vec: two or more VCTRs that lie on the same line or are parallel to each other.

Equal VCTR: two vectors that have the same magnitude and direction, regardless of their starting point.

Displacement vector: This represents the change of position from one point to another.

Negative vector: Which have the same size as the original but "V" point in the opposite direction.

Parallel Vec: They have the same or parallel direction, but can have different magnitudes.

Orthogonal Vec: These are perpendicular to each other, meaning their dot product is zero.

Linearly independent VCTRs: A collection of VCTRs in which none of the vectors can be expressed as a linear combination of the others.

Basis Vec: These form the basis of a vector space, meaning that every VCTR in that space can be expressed as a linear combination of the basis vectors.

Eigenvectors: Which associated with a linear transformation that only changes by a scalar factor when the transformation is applied.

Vectors support various operations to perform manipulations and calculations. Here are some common  operations:

1)Addition: When two vectors are added, their components are added together. In component form, if A = (A₁, A₂, A₃) and B = (B₁, B₂, B₃), then the sum of  A and B, denoted as A + B, is (A₁ + B₁, A₂ + B₂, A₃ + B₃).

Triangle Law: If two sides of a triangle are represented by two continuous vectors ( A and B), then the third side of the triangle in the opposite direction shows the resultant of the two vectors, which is ( C).

 
 The addition is commutative.

⇒ A + B = B + A

The addition is associative.

⇒ A+(B+C) = (A+B)+C

If all sides of a polygon are continuous vectors, then the sum of the vectors of all sides is zero.

Polygon Method: We use this method when we need to add more than two vectors. This is a development of the vector addition of the triangular law. If different vectors in magnitude and direction can be represented by the sides of the polygon taken in the same order, then their resultant magnitude and direction are the closed side of the polygon, viewed from the other direction.

2)Scalar multiplication: A vector can be multiplied by a scalar value, which is a single numerical value. The scalar multiplies each component of the vector. For example, if A = (A₁, A₂, A₃) is a vector and k is a scalar, then the inner product is kA (kA₁, kA₂, kA₃).

3)Dot Product: Dot product is an operation that combines two vectors and produces a scalar. It is calculated by multiplying the respective components of the vectors and adding the results. In component form, if A = (A₁, A₂, A₃) and B = (B₁, B₂, B₃), then the scalar product of A and B, denoted AB, is A₁B₁ + A₂B₂ + A₃B₃. The dot product is useful for finding angles between vectors and determining whether they are orthogonal (perpendicular) to each other.

4)Cross product: A cross product is an operation that combines two vectors and creates a new one that is perpendicular to both of the original ones. The cross-product is defined only in three dimensions. In component form, if A = (A₁, A₂, A₃) and B = (B₁, B₂, B₃), then the cross product of A and B, denoted as A × B, is given by, (A₂B₃ - A₃B₂, LA₃B₁ - A₁B₃, A₁B₂ - A₂B₁). The cross-product is useful for determining the direction of a perpendicular vector and calculating areas and volumes.

5) Subtraction: It is similar to addition, but instead of adding the corresponding components, we subtract them. In component form, if A = (A₁, A₂, A₃) and B = (B₁, B₂, B₃), then the difference between A and B, denoted as A - B, (A₁ - B₁, A₂ - B₂, A₃ - B₃).

Uses:

Vectors have numerous uses in everyday life often in ways we do not indeed realize. Here are a few common employments:

Navigation:

They are broadly utilized in route frameworks such as GPS (Global Positioning System). GPS uses vectors to calculate distances, directions, and routes from one place to another.

Move and Work out:

They play an imperative part in describing and analyzing motion. In sports, they help to decide the speed, increasing speed, and direction of moving objects such as balls, athletes, and vehicles.

Architecture and Engineering:

Architects and engineers utilize vectors to plan and construct structures. They are utilized to represent forces acting on a structure, such as tensile, compressive, and shear forces. They help decide the stability and balance of buildings and bridges.

Computer Design:

They are the backbone of computer design and animation. They are utilized to represent and manipulate objects in three-dimensional (3D) space, giving reasonable representation and development of objects in video recreations, motion pictures, and simulations.

Electrical Designing:

They are utilized to represent and analyze electrical quantities such as voltage, current, and impedance. They are fundamental for understanding the behavior of electrical circuits and designing electrical systems.

Fluid Dynamics:

They are utilized to describe the flow of liquids such as air and water. They offer assistance in analyzing the speed, weight, and heading of liquid streams in applications such as aerodynamics, liquid flow, and weather forecasting.

Economy and Finance:

They are utilized in financial matters and funds to model and analyze economic variables such as supply, demand, and price. They are used to represent quantities and relationships between different factors in economic models.

Optimization and Logistics:

They are utilized in optimization problems such as finding the shortest way or the foremost proficient way. They offer assistance in deciding the direction and magnitude of changes required to optimize the system, making them useful for coordination and supply chain management.

Genetics and Molecular Biology:

They are utilized in molecular biology to transport and control DNA. In hereditary designing, they are utilized as vehicles to exchange genetic material into cells for different purposes, such as gene treatment or the generation of hereditarily adjusted living beings.