Complex Numbers and Vectors

A complex number is a number that can be expressed in the form z = x + iy, where x and y are real numbers, and 'i' is the imaginary unit, satisfying i² = -1. 'x' is called the real part (Re(z)) and 'y' is called the imaginary part (Im(z)).

The complex conjugate of z = x + iy is denoted as z̄ (read as 'z bar') and is given by z̄ = x - iy. The product of a complex number and its conjugate is always a real number: z * z̄ = x² + y².

Given z₁ = x₁ + iy₁ and z₂ = x₂ + iy₂:\n- Addition: z₁ + z₂ = (x₁ + x₂) + i(y₁ + y₂)\n- Subtraction: z₁ - z₂ = (x₁ - x₂) + i(y₁ - y₂)\n- Multiplication: z₁z₂ = (x₁x₂ - y₁y₂) + i(x₁y₂ + x₂y₁)\n- Division: z₁/z₂ = (z₁ * z̄₂)/(z₂ * z̄₂). Multiply the numerator and denominator by the conjugate of the denominator.

A complex number z = x + iy can be represented in polar form as z = r(cos θ + i sin θ), where 'r' is the modulus (distance from the origin to z on the Argand diagram) and 'θ' is the argument (angle measured anticlockwise from the positive real axis to the line segment connecting the origin to z). r = |z| = √(x² + y²) and θ = arg(z). The principal argument is typically in the range -π < θ ≤ π.

Using Euler's formula, e^(iθ) = cos θ + i sin θ, the polar form can be written as z = re^(iθ). This form is particularly useful for multiplication, division, powers, and roots of complex numbers.

For any integer n, De Moivre's Theorem states that (r(cos θ + i sin θ))^n = r^n(cos(nθ) + i sin(nθ)). This theorem is fundamental for finding powers and roots of complex numbers.

A vector is a quantity that has both magnitude (size) and direction. Examples include displacement, velocity, and force. A scalar quantity, such as mass or temperature, has only magnitude.

Vectors can be represented in component form, column vector form, or using unit vector notation:\n- Component form: v = (x, y) for 2D, v = (x, y, z) for 3D.\n- Column vector: v = [x, y] for 2D, v = [x, y, z] for 3D.\n- Unit vector notation: v = xi + yj for 2D, v = xi + yj + zk for 3D, where i, j, k are standard unit vectors along the x, y, z axes respectively.

The magnitude (or length) of a vector v = (x, y, z) is denoted by |v| and is calculated using the Pythagorean theorem.

A unit vector is a vector with a magnitude of 1. The unit vector in the same direction as a non-zero vector v is denoted by v̂ and is found by dividing the vector by its magnitude.

A position vector represents the position of a point relative to the origin. If P is a point (x, y, z), its position vector is OP = (x, y, z). The vector from point A to point B is given by AB = OB - OA, where OA and OB are the position vectors of A and B respectively.

Given a = (a₁, a₂, a₃) and b = (b₁, b₂, b₃):\n- Addition: a + b = (a₁ + b₁, a₂ + b₂, a₃ + b₃)\n- Subtraction: a - b = (a₁ - b₁, a₂ - b₂, a₃ - b₃)\n- Scalar Multiplication: For a scalar k, k * a = (ka₁, ka₂, ka₃).

The dot product of two vectors a and b is a scalar quantity defined as a ⋅ b = |a||b|cos θ, where θ is the angle between the vectors. In component form, a ⋅ b = a₁b₁ + a₂b₂ + a₃b₃. If a ⋅ b = 0 for non-zero vectors, then a and b are perpendicular.

The cross product of two 3D vectors a and b is a vector quantity defined as a × b = |a||b|sin θ n̂, where n̂ is a unit vector perpendicular to both a and b, determined by the right-hand rule. The magnitude |a × b| represents the area of the parallelogram formed by a and b. In component form:\na × b = (a₂b₃ - a₃b₂, a₃b₁ - a₁b₃, a₁b₂ - a₂b₁). If a × b = 0 for non-zero vectors, then a and b are parallel.