Engineering Mathematics 1 – Complex Numbers, Vectors, and Matrices (Chs 1-3)

A set of practice flashcards covering key concepts from complex numbers, Argand diagrams, De Moivre, roots, vectors, and introductory matrices, based on the provided lecture notes.

What does the Fundamental Theorem of Algebra state?

Every polynomial equation with complex coefficients of degree n has exactly n roots in the complex numbers, counting multiplicities.

What is a complex number?

A complex number is of the form z = x + yi where x and y are real numbers and i^2 = −1.

What are the real part and imaginary part of a complex number z = x + yi?

Re(z) = x and Im(z) = y.

How do you determine if two complex numbers z and w are equal?

They are equal iff Re(z) = Re(w) and Im(z) = Im(w).

What is the Argand diagram?

A geometric representation of complex numbers in the plane, identifying z = x + yi with the point (x, y) (and viewing z as a vector from the origin).

How is the modulus of a complex number z = x + yi defined?

|z| = sqrt(x^2 + y^2).

What is the argument of a complex number z, and what is the principal argument?

arg(z) is the angle θ such that x = |z| cos θ and y = |z| sin θ; the principal argument is the angle chosen in (−π, π].

What is the polar form of a complex number z?

z = r(cos θ + i sin θ), where r = |z| and θ is an argument of z.

What is the exponential form of z in terms of r and θ?

z = r e^{i θ}.

What is the conjugate of z = x + yi?

z* = x − yi.

What is the conjugate of a complex number in polar/exponential form?

If z = r(cos θ + i sin θ) then z* = r(cos(−θ) + i sin(−θ)) = r e^{−i θ}.

What are the modulus and argument of the product z1 z2?

|z1 z2| = |z1| |z2| and arg(z1 z2) = arg(z1) + arg(z2) (mod 2π).

How do you divide two complex numbers z1 and z2 using the conjugate method?

z1 / z2 = z1 z2* / |z2|^2, where z2* is the conjugate of z2.

State De Moivre's Theorem.

(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ) for every integer n.

What are the nth roots of a nonzero complex number z = r(cos θ + i sin θ)?

z_k = r^{1/n} [cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)], k = 0,1,…,n−1.

What are the nth roots of unity?

The n distinct complex numbers z_k = cos(2πk/n) + i sin(2πk/n), k = 0,…,n−1, solving z^n = 1.

How is the product of two complex numbers expressed in polar form?

If z1 = r1(cos θ1 + i sin θ1) and z2 = r2(cos θ2 + i sin θ2), then z1 z2 = r1 r2 [cos(θ1 + θ2) + i sin(θ1 + θ2)].

What is the dot product of two vectors u and v in R^2?

u · v = u1 v1 + u2 v2.

How do you compute the angle between two vectors using the dot product?

cos φ = (u · v) / (||u|| ||v||).

What is the cross product and its basic properties in R^3?

u × v is a vector perpendicular to both u and v; |u × v| = ||u|| ||v|| sin φ; direction is given by the right-hand rule; u × v = −(v × u).

What is the work done by a constant force F moving through displacement D?

W = F · D.

What is the projection of vector u onto vector v?

proj_v(u) = (u · v / ||v||^2) v.

How do you compute the distance between two points P and Q in 3-space?

d = ||PQ|| = sqrt((x2 − x1)^2 + (y2 − y1)^2 + (z2 − z1)^2).

What is the parametric equation of a line through r0 with direction v?

r(t) = r0 + t v, t ∈ ℝ.

What is the Cartesian form of a line given a point r0 and direction v?

x = x0 + t v1, y = y0 + t v2, z = z0 + t v3; equivalently (x − x0)/v1 = (y − y0)/v2 = (z − z0)/v3 (where components are nonzero).

What is the equation of a plane with normal n = (a, b, c) through P0 = (x0, y0, z0)?

(n) · (P − P0) = 0 → a(x − x0) + b(y − y0) + c(z − z0) = 0; equivalently ax + by + cz = d with d = ax0 + by0 + cz0.

How do you compute the distance from a point to a plane with normal n?

Distance = |(P − S) · n| / ||n||, where S is a point on the plane.

What is the area of a triangle ABC in terms of vectors AB and AC?

Area = 1/2 ||AB × AC||.

What is the determinant of a 2×2 matrix [a b; c d] and when is it invertible?

det = ad − bc; the matrix is invertible if ad − bc ≠ 0.

What is the inverse of a 2×2 matrix A = [a b; c d] when invertible?

A^{-1} = (1/(ad − bc)) [d −b; −c a].

What is the adjoint (adj(A)) and its relation to A and det(A)?

adj(A) is the transpose of the cofactor matrix; A adj(A) = det(A) I, and if det(A) ≠ 0 then A^{-1} = (1/det(A)) adj(A).