عمليات المتجهات والفيزياء

Vector Operations

• Vectors: Quantities with both magnitude (size) and direction.
• Definitions: a, b, and c are vectors, 2a + 5b – 3c (represents a linear combination).
• Coordinates: Vectors can be represented using coordinates (e.g., (2, 4, -3)).
• Magnitude: Size of a vector calculated using the Pythagorean Theorem or formula depending on the dimensions.
• Components (or projections): Values which define vector's direction and strength, in coordinate planes, which may involve axes like i, j, k.
• Addition: Vector addition (e.g., (2, 5) + (-3, 2) = (-1, 7)).
• Scalar multiplication: Multiplying a vector by a constant (scalar).

Finding Distance Between Points

• Finding distance between two points (e.g., determining the distance between two balloons (20, 25, 30) and (-29, 15, 10)).

Applications of Vectors in Physics

• Velocity and Displacement: Vectors used to represent velocity and displacement. Given information for a football with speeds.
• Finding Resultant Velocity: Calculating the resultant velocity which combines different velocities.

Dot Product

• Definition: Dot product of two vectors (u, v) involves finding the product of their corresponding components and summing them up which may be used to identify if they are orthogonal or not (i.e., at 90 degrees).
• Orthogonal vectors: Vectors that form a 90-degree angle, indicated by a null dot product.

Cross Product

• Definition: A vector operation resulting in a new vector perpendicular to both input vectors.

Coordinate Systems

• Coordinate planes: Coordinate systems used in 3D space.
• 3D Cartesian coordinate systems: Use coordinates represented by x, y, z and are involved in the calculations to find distances along or between coordinates such as in the problem of balloon coordinates.