maths.free › Precalculus › Vectors

Vectors

Magnitude, direction, components, and adding vectors head to tail.

A vector is a quantity with magnitude and direction — displacement, velocity, force. In components it is just a column of numbers, adding entry by entry; its length comes from Pythagoras. Linear algebra picks up here.

مثال عملي: [[3],[4]]

Linear-Algebra · Matrices

[[3],[4]]

\left[\begin{matrix}3\\4\end{matrix}\right]

خطوة خطوة

  1. \left[\begin{matrix}3\\4\end{matrix}\right]

    A 2×1 matrix.

  2. \operatorname{rank} A = 1

    Rank (number of independent rows).

  3. A^T = \left[\begin{matrix}3 & 4\end{matrix}\right]

    Transpose.

كشفت الإجابة
\left[\begin{matrix}3\\4\end{matrix}\right]

Symbols used here

A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}
matrix
A rectangular array of numbers; a linear map.
\leq,\ \geq
less/greater than or equal
Inequalities that allow equality; < and > exclude it.
|x|
absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.
\binom{n}{k}
binomial coefficient, "n choose k"
Number of k-element subsets of n things: n!/(k!(n−k)!).
\pi
pi
Ratio of a circle's circumference to its diameter, 3.14159…
e
Euler's number
2.71828…, the base whose exponential is its own derivative.
i
imaginary unit
i² = −1.
\theta
theta
The usual name for an angle.
\log_b x,\ \ln x
logarithm, natural log
The exponent b must be raised to for x; ln uses base e.
\mathbf{v},\ \vec{v}
vector
A quantity with magnitude and direction; a column of numbers.

How to: Vectors

  1. A 2×1 matrix.
  2. Rank (number of independent rows).
  3. Transpose.

Questions people ask

What is a function, really?

A rule that assigns exactly one output to each input. The vertical-line test on a graph is the same idea: no input may have two outputs.

Why do we need complex numbers?

Because x² + 1 = 0 has no real solution, and allowing one new number i with i² = −1 makes every polynomial equation solvable. They then turn out to describe rotation, waves and alternating current more naturally than real numbers do.

جرّب نفسك

أكثر في Precalculus

Complex numbersPolynomial functionsRational functionsSequences and seriesThe binomial theoremConic sectionsExponential and logarithmic functionsPolynomial division and the remainder theoremParametric equations and polar coordinates