Cuba 12th Grade Math Final Exam 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).

Selections of k elements from n without regard to order, counted by C(n,k) = n!/(k!(n−k)!); used when only the group matters.

Ordered arrangements of all or part of a set of elements; the number of orderings of n distinct items is n!, and repeated items require division by their factorials.

Probability of A given that B has occurred, P(A|B) = P(A∩B)/P(B), defined when B has non-zero probability.

Two events are independent when P(A∩B) = P(A)·P(B), so knowing one occurred does not change the probability of the other.

Formula reversing a conditional probability, P(A|B) = P(B|A)P(A)/P(B), with the denominator expanded over a partition of the sample space.

Graph whose branches represent successive stages of an experiment with probabilities on the edges; path products give intersections and sums give totals.

Model for the number of successes in n independent trials with constant success probability p; probabilities come from C(n,k)pᵏ(1−p)ⁿ⁻ᵏ, with mean np.

Continuous symmetric bell-shaped distribution fixed by mean μ and standard deviation σ; probabilities are areas under the curve found by standardising to N(0,1).

Change of variable z = (x − μ)/σ that converts any normal variable into the standard normal N(0,1), allowing use of a single table.

Use of a normal with mean np and variance np(1−p) to estimate binomial probabilities when n is large and p is not extreme, applying a continuity correction.

Range of plausible values for an unknown population parameter, built from a sample statistic plus or minus a margin set by the confidence level.

Standard deviation of the sample mean, equal to σ/√n, or to s/√n when the population standard deviation is unknown; it shrinks as the sample grows.

For independent identically distributed observations with finite variance, the standardized sample mean approaches a normal distribution as sample size increases. The approximation quality depends on population shape and sample size; it is not guaranteed for every arbitrary population.

Procedure comparing a null hypothesis about a parameter with sample data, deciding rejection when the standardised statistic falls in a critical region.

The arithmetic average of a data set, found by summing all values and dividing by the count; sensitive to extreme values or outliers.

The middle value of an ordered data set, or the average of the two middle values when the count is even; resistant to outliers compared with the mean.

A measure of the typical distance of data values from the mean, equal to the square root of the variance; larger values indicate more spread.

A symmetric, bell-shaped probability distribution defined by its mean and standard deviation; roughly 68% of values fall within one standard deviation of the mean.