vectors 2d 3d

Vectors are mathematical objects representing quantities with both magnitude (size) and direction, unlike scalars which have only magnitude. In Cambridge A-Level, vectors are denoted using bold type (a) or underlining (ā) or with column notation.

Key Definitions:

• Position vector: Vector from origin O to point P, written OP or r
• Displacement vector: Vector from point A to point B, written AB = b - a
• Magnitude: Length of vector a = (x, y, z) calculated as |a| = √(x² + y² + z²)
• Unit vector: Vector with magnitude 1, found by â = a/|a|
• Base vectors: i, j, k represent unit vectors along x, y, z axes respectively

Essential Formulae:

Vector operations:

• Addition: a + b = (a₁ + b₁, a₂ + b₂, a₃ + b₃)
• Scalar multiplication: λa = (λa₁, λa₂, λa₃)
• Scalar (dot) product: a · b = a₁b₁ + a₂b₂ + a₃b₃ = |a||b|cos θ

Key properties:

• Parallel vectors: a = λb for some scalar λ
• Perpendicular vectors: a · b = 0
• Angle between vectors: cos θ = (a · b)/(|a||b|)

Mnemonic: "MUD" - Magnitude using Distance formula, Unit vector from Division, Dot product for angles.

Cambridge Key Point: Always express final answers in the form requested—column vector, component form, or i, j, k notation as specified in the question.

Real-World Context: Vectors model physical phenomena where direction matters—velocity, force, displacement. Imagine GPS navigation: your position is a position vector from a reference point, while directions like "5 km northeast" are displacement vectors.

2D Vector Applications: Aircraft navigation uses 2D vectors. If wind velocity is 30i + 40j km/h (where i = east, j = north), pilots must add their airspeed vector to find ground velocity. A plane flying at 200i km/h faces resultant velocity of 230i + 40j km/h—they actually travel northeast, not due east!

3D Vector Applications: Structural engineers use 3D vectors for forces. A cable supporting a bridge component might exert force F = 100i - 50j + 200k Newtons. The magnitude |F| = √(100² + 50² + 200²) ≈ 229N represents total tension.

Scalar Product in Practice: Work done by force is F · d (force dot displacement). If you push a lawnmower with force 50N at 30° to horizontal over 10m, work = 50 × 10 × cos(30°) ≈ 433J. The dot product automatically accounts for the angle—only the component of force in the direction of motion does work.

Perpendicular Vectors Analogy: Think of perpendicular vectors like perpendicular streets: walking north doesn't get you any closer to an eastern destination. When a · b = 0, one vector contributes nothing in the direction of the other—mathematically "orthogonal."

Visualization Tip: Sketch vectors as arrows. Vector addition uses the triangle rule or parallelogram rule—drawing these helps prevent sign errors.