عمليات المتجهات وتطبيقاتها

Vector Operations and Applications

• Vector Addition/Subtraction: Vectors are added/subtracted component-wise. For example, if u = (a, b, c) and v = (d, e, f), then u + v = (a + d, b + e, c + f) and u - v = (a - d, b - e, c - f).

• Scalar Multiplication: Multiplying a vector by a scalar (number) multiplies each component of the vector by that scalar. For example, if u = (a, b, c) and k is a scalar, then ku = (ka, kb, kc).

• Dot Product: The dot product of two vectors u and v, denoted as u ⋅ v, is calculated by multiplying corresponding components and summing the results. If u = (u₁, u₂, u₃) and v = (v₁, v₂, v₃), then u ⋅ v = u₁v₁ + u₂v₂ + u₃v₃. The dot product can determine if two vectors are orthogonal (perpendicular).

• Cross Product: The cross product of two vectors u and v, denoted as u × v, is a vector perpendicular to both u and v. Its magnitude is given by |u × v| = |u| |v| sin θ, where θ is the angle between u and v. Its direction is found using the right-hand rule.

• Vector Equations: Equations can represent relationships between vectors, such as u = (2-3, 1) t + (1, 4) where t is a scalar.

• Parallelogram/Triangle Law of vectors: Graphical methods for vector addition.

• Applications: Vectors are used to represent and solve many real-world problems in physics, engineering, and computer graphics (e.g., forces, displacements, velocities). Example applications are: finding magnitudes, directions, determining if vectors are orthogonal (perpendicular) and calculating angles between vectors.

Specific Examples of Vector Operations and Problems

• Given vectors u, v, and t, find particular values: u+v, u-v, ku, u ⋅ v, u × v, or determine whether the vectors are orthogonal
• Example values of vectors: u = (-1, -9, 2), v = (3, -2, 6)
• Examples: Find the angles between vectors; find vectors related to the dot product
• Applications determining geometric shapes such as parallelograms, triangles.
• Special cases, such as determining if points are collinear.