Mathematics: Vectors and Probability Concepts

Mathematics

• Vectors:
  • Vectors are quantities that have both magnitude (size) and direction.
  • Represented graphically by arrows, with the length of the arrow representing the magnitude and the direction of the arrow representing the direction.
  • Vectors can be added, subtracted, and scaled. Vector addition is the parallelogram law.
  • The dot product of two vectors is a scalar quantity. The dot product measures the projection of one vector onto another.
  • The cross product of two vectors is a vector quantity. The cross product is perpendicular to both of the original vectors.
  • Unit vectors are vectors with a magnitude of 1. They often have a directional component.
  • Vector components can determine a vector's position in space (x, y, and possibly z coordinates for 3D).
  • Applications of vectors in physics include forces, velocities, and displacements.

Probability

• Probability:
  • Probability measures the likelihood of an event occurring. Ranges from 0 (impossible) to 1 (certain).
  • Basic concepts include sample space, events, and outcomes.
  • Conditional probability is the probability of an event occurring given that another event has already occurred.
  • Independent events do not influence each other's probability.
  • Expected value represents the average outcome of a random variable.
  • Combinations and permutations are useful for calculating probabilities when dealing with selecting items from a set.
  • Many probability distributions exist, including the binomial, normal, and Poisson distributions.

Trigonometry 1 & 2

• Trigonometry 1:

  • Focus primarily on trigonometric ratios (sin, cos, tan) for right-angled triangles.
  • Application includes solving for sides and angles in right-angled triangles.
  • Trigonometric identities relate different trigonometric functions.
  • The unit circle helps visualize trigonometric functions and their periodicity.
  • Identities like sin²θ + cos²θ = 1 are foundational.

• Trigonometry 2:

  • Builds upon Trigonometry 1, often expanding to non-right angled triangles using the Law of Sines and the Law of Cosines.
  • Covers applications to more complex geometric problems.
  • Introduces angles of elevation and depression often encountered in surveying and related applications.
  • May involve solving for sides, angles and area of any triangle.
  • Polar coordinates use angles and distances from the origin to represent points.

Logarithms

• Logarithms:
  • Logarithms are the inverse operation to exponentiation.
  • For example, log_b(x) = y means b^y = x.
  • Common logarithms use base 10 (log). Natural logarithms use base 'e'.
  • Logarithms can be useful for simplifying calculations involving very large or very small numbers, such as in scientific contexts.
  • Logarithm rules, such as the logarithm of a product, logarithm of a quotient, logarithm of a power, and change of base, are often important. For example log(ab) = log(a) + log(b)
  • Inverse functions, including logarithms, play a crucial role in finding solutions to equations.