Přijímačky VŠE Matematika key terms

Non-zero vector parallel to a line; together with one point of the line it determines the line completely in vector, parametric and continuous forms.

Non-zero vector perpendicular to a plane; its components are exactly the coefficients of x, y and z in the general equation of the plane.

Operation on two vectors giving the number |u||v|cos θ; zero result means the vectors are perpendicular, and it yields the angle between them.

Operation on two vectors in space giving a third vector perpendicular to both, with modulus |u||v|sin θ; used to build normals and to compute areas of parallelograms.

Scalar u·(v×w), equal to the determinant of the three vectors as rows; it gives the signed volume of the parallelepiped and vanishes when they are coplanar.

Three or more vectors lying parallel to a single plane; three of them are coplanar exactly when their mixed product is zero.

Form x = a₁ + t·v₁, y = a₂ + t·v₂, z = a₃ + t·v₃, where (a₁,a₂,a₃) is a point, v the direction vector and t the real parameter.

Equation Ax + By + Cz + D = 0, where (A,B,C) is a normal vector; substituting a point of the plane fixes the constant D.

Classification of two lines as coincident, parallel, intersecting or skew, decided by comparing their direction vectors and checking whether a vector joining points lies in their span.

Two lines in space that are neither parallel nor intersecting, so they lie in different parallel planes and share no point.

For a nonzero line direction v and plane normal n, the non-obtuse angle alpha satisfies sin(alpha)=abs(v·n)/(|v||n|). It is complementary to the acute or right angle between the line direction and the normal.

Length of the perpendicular segment from the point to the plane, computed as |Ax₀+By₀+Cz₀+D| divided by the modulus of the normal vector (A,B,C).

A rule that assigns to each input value in its domain exactly one output value, often written f(x); passing the vertical line test distinguishes functions from general relations.

The domain is the set of all valid input values for which a function is defined; the range is the set of all output values the function actually produces.

A function that reverses another function, so f and g are inverses when f(g(x)) = x and g(f(x)) = x; it exists only where the original function is one-to-one.

A new function formed by applying one function to the output of another, written (f ∘ g)(x) = f(g(x)); order matters because composition is generally not commutative.

A function of the form f(x) = ax² + bx + c with a ≠ 0, whose graph is a parabola opening upward when a > 0 and downward when a < 0.

The highest or lowest point on a parabola, located at x = -b/(2a) for f(x) = ax² + bx + c; it represents the maximum or minimum value of the quadratic function.

The quantity b² - 4ac from the quadratic formula; its sign indicates whether a quadratic equation has two real roots, one repeated real root, or two complex conjugate roots.

A technique that rewrites a quadratic expression as a perfect square plus a constant, converting ax² + bx + c into vertex form and enabling solution of quadratic equations.

A function of the form f(x) = a·bˣ with b > 0 and b ≠ 1; it models constant multiplicative growth or decay and has a horizontal asymptote at y = 0.

The exponent to which a base must be raised to produce a given number: log_b(x) = y means bʸ = x. It is the inverse relationship of exponentiation.

A number of the form a + bi where a and b are real and i² = -1; complex numbers extend the real line so that every nonconstant polynomial equation with complex coefficients has a complex root.

Every nonconstant polynomial with complex coefficients has at least one complex root, so a polynomial of degree n has exactly n roots counted with multiplicity.

An expression of the form aₙxⁿ + ... + a₁x + a₀ with nonnegative integer exponents and real coefficients; its degree is the largest exponent with nonzero coefficient.

A function written as the ratio of two polynomials, P(x)/Q(x); it may have vertical asymptotes where Q(x) = 0 and horizontal or slant asymptotes determined by degrees.

A line that a curve approaches arbitrarily closely as the input or output grows without bound, without necessarily ever touching it; common types are vertical, horizontal, and slant.

A function in which distinct inputs always produce distinct outputs, so f(a) = f(b) implies a = b; such functions pass the horizontal line test and possess an inverse function.

A function such as sine, cosine, or tangent defined from ratios of sides in a right triangle or from coordinates on the unit circle; these functions are periodic.

The circle of radius 1 centered at the origin; a point at angle θ on it has coordinates (cos θ, sin θ), which defines the trigonometric functions for all real angles.