Vectors in Science

Vectors in Science

• Vectors are used in mathematics, physics, and engineering.
• They describe velocity, acceleration, forces, reactions, and couples in moving objects.

Definition of a Vector

• A vector has both magnitude and direction.
• Geometrically, a vector is a directed line segment. The length represents magnitude, and the direction goes from the tail to the head.

Classification of Vectors

• Free Vector: Action is not confined to a specific line. Magnitude and direction remain the same regardless of the application point (e.g., force applied at different points on a box).
• Sliding Vector: A unique line of action exists, but the application point can vary along that line without altering the effect on the whole system (e.g., force on a rigid body).
• Fixed Vector: A specific application point is needed. The force's effect depends on both the application point and other factors (e.g., force on a deformable object).

Notation

• Vectors are denoted by capital letters with an arrow on top (e.g., A→, B→).
• Unit vectors are denoted by lowercase letters with a circumflex (e.g., û).

Representation of Vectors

In a Cartesian Plane

• A vector in a Cartesian plane (2D) is represented as oA→ = (ax, ay), where (ax, ay) are coordinates of point A relative to origin O.

• Alternatively, oA→ = axî + ayĵ, where î and ĵ are unit vectors along the x and y axes, respectively.

• Components (ax and ay) can be calculated based on the vector's length (‖oA→‖) and angle θ with the x-axis.

• ax = ‖oA→‖ cos θ

• ay = ‖oA→‖ sin θ

• tan θ = ay/ax

• Vector length calculation: ‖oA→‖ = √(ax² + ay²)

In Three-Dimensional Space

• Vectors in 3D space are represented by triples of scalar components (ax, ay, az).

• oA→ = axî + ayĵ + azk̂, where k̂ is the unit vector along the z-axis.

• Component calculations (az, ax, ay) depend on the angle θ with the z-axis and angles φ with x-axis (to determine x and y components)

• Unit vector calculation: û = u→ / ‖u→‖