Vector Calculator

**Vector operations:** For vectors A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃): **Addition:** A + B = (a₁ + b₁, a₂ + b₂, a₃ + b₃) **Subtraction:** A - B = (a₁ - b₁, a₂ - b₂, a₃ - b₃) **Scalar multiplication:** k × A = (ka₁, ka₂, ka₃) **Magnitude (length):** |A| = √(a₁² + a₂² + a₃²) **Unit vector:** Â = A / |A| = (a₁/|A|, a₂/|A|, a₃/|A|) **Worked example: A = (3, 4, 0), B = (1, 2, 2)** |A| = √(9 + 16 + 0) = 5. |B| = √(1 + 4 + 4) = 3. Unit vector A: (3/5, 4/5, 0) = (0.6, 0.8, 0). Unit vector B: (1/3, 2/3, 2/3) ≈ (0.333, 0.667, 0.667). A + B = (4, 6, 2). A - B = (2, 2, -2). B - A = (-2, -2, 2). **Distance between two points:** distance = |A - B| or |B - A|. For points A and B: d = √((b₁-a₁)² + (b₂-a₂)² + (b₃-a₃)²). For (3,4,0) and (1,2,2): d = √(4 + 4 + 4) = √12 ≈ 3.46. **Direction angle (2D vector):** For A = (a₁, a₂): angle from x-axis: θ = arctan(a₂/a₁). For A = (3, 4): θ = arctan(4/3) ≈ 53.13°. **Direction in 3D (spherical):** For A = (a₁, a₂, a₃): - Polar angle (from z-axis): φ = arccos(a₃/|A|). - Azimuthal angle (in xy-plane): θ = arctan(a₂/a₁). **Vector arithmetic properties:** | Property | Statement | |---|---| | Commutative (add) | A + B = B + A | | Associative (add) | (A + B) + C = A + (B + C) | | Distributive | k(A + B) = kA + kB | | Identity | A + 0 = A | | Inverse | A + (-A) = 0 | Note: vector cross product is NOT commutative (A × B = -(B × A)). **Linear combination:** Any vector v in space can be expressed as combination of basis vectors: v = a₁ × ê₁ + a₂ × ê₂ + a₃ × ê₃ Where ê = unit vectors along axes. **Decomposition:** Vector v can be decomposed into components along and perpendicular to a reference vector u: - Parallel: (v · û) × û. - Perpendicular: v - parallel. Used in: physics (forces parallel and perpendicular to inclines), optics (incident and reflected components). **Common 2D vector applications:** | Use | Example | |---|---| | Position | (x, y) coordinates | | Velocity | (vₓ, vᵧ) m/s | | Force | (Fₓ, Fᵧ) N | | Displacement | from A to B | | Direction | unit vector | **Common 3D vector applications:** | Use | Example | |---|---| | Position | (x, y, z) in space | | Velocity | (vₓ, vᵧ, v_z) | | Force | 3D force vector | | Angular momentum | L = r × p | | Normal vector | surface orientation | **Resultant vector:** For multiple vectors added: sum component-wise. For forces F₁, F₂, F₃ acting on object: F_resultant = F₁ + F₂ + F₃. If F_resultant = 0: object in equilibrium. **Worked example: forces on object** F₁ = (10, 0) N, F₂ = (-5, 8) N, F₃ = (-5, -8) N. Sum: F = (10 - 5 - 5, 0 + 8 - 8) = (0, 0). Object is in equilibrium. **Vector magnitudes for common physics:** - Speed = |velocity|. - Force magnitude = |force vector|. - Distance = |displacement vector|. **Unit vector intuition:** Unit vector preserves direction, normalizes magnitude to 1. For (3, 4, 0) with magnitude 5: unit (3/5, 4/5, 0) = (0.6, 0.8, 0). Magnitude check: √(0.36 + 0.64 + 0) = √1 = 1 ✓. Used to specify direction without magnitude (e.g., wind from northeast direction). **Vector triangle:** For vector triangle: A + B + C = 0. Useful in equilibrium problems, position triangles. **Worked example: walking** Walk 5 m east, then 4 m north. Final position relative to start? Displacement 1: (5, 0). Displacement 2: (0, 4). Total: (5, 4). Distance: √(25 + 16) = √41 ≈ 6.4 m. Direction: arctan(4/5) ≈ 38.7° from east. **Vector projection:** Projection of A onto B: proj_B(A) = ((A · B) / |B|²) × B. Length: (A · B) / |B|. Used for: physics (work calculation), graphics (shadow computation). **Cross product** (3D only): A × B = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁). Result perpendicular to both A and B. **Dot product:** A · B = a₁b₁ + a₂b₂ + a₃b₃. Scalar result. Measures alignment. **Common operations summary:** | Operation | Result | Use | |---|---|---| | Add | Vector | Resultant of forces | | Subtract | Vector | Displacement A→B | | Scalar × | Vector | Scaling | | Magnitude | Scalar | Length | | Unit vector | Vector | Direction only | | Dot product | Scalar | Alignment, work | | Cross product (3D) | Vector | Torque, perpendicular | **Common applications:** - **Physics**: force, velocity, acceleration analysis. - **Engineering**: structural analysis, beam loading. - **Computer graphics**: 3D positions, transformations. - **Robotics**: end-effector position, motion planning. - **GPS/Navigation**: positions, course corrections. - **Aerospace**: spacecraft trajectory, attitudes. - **Game development**: character movement, physics. - **Animation**: keyframe interpolation. **Pitfalls:** - **Vector vs scalar**: keep distinct. - **Component order**: (x, y, z) standard. - **Magnitude is always positive**. - **Unit vector requires division by non-zero magnitude**. - **Cross product anti-commutativity**: A × B = -(B × A). - **For 2D**: cross product gives scalar (z-component only). **Vector in coordinate systems:** - **Cartesian** (x, y, z): most common. - **Cylindrical** (ρ, φ, z): rotational symmetry. - **Spherical** (r, θ, φ): radial symmetry. Convert between as needed for specific problems. **Software:** - **Python (NumPy)**: extensive vector operations. - **MATLAB**: built-in. - **C++ Eigen**: high-performance. - **JavaScript THREE.js**: 3D graphics. - **Mathematica**: symbolic vector calculus. **Educational notes:** Vectors introduced in: - High school physics: forces, velocities. - Pre-calculus: 2D and 3D vectors. - Calculus III: vector calculus. - Linear algebra: vector spaces. Foundation for nearly all quantitative sciences. **Pitfalls (continued):** - **For magnitude**: take positive square root. - **For direction angle**: account for quadrant (arctan ambiguous). - **For unit vector of zero**: undefined. - **Vector addition triangle**: head-to-tail diagram. - **For 2D representation**: drop z = 0 from 3D formulas.