Vectors 2:


Nat 5 info:

A vector is a quantity which has both direction and magnitude.

  • The magnitude of a vector is its size or length.
  • A directed line segment from A to B is defined as LaTeX: $\overrightarrow {AB} $.
  • A vector or force can also be defined by a lower-case letter in bold.
  • Vectors are equal if they have the same direction and magnitude.
  • When a vector has its direction reversed it is negative e.g. a becomes -a.
  • A vector can be multiplied by a scalar e.g. doubling a gives 2a.
  • Vectors can be added by joining one to the end of another.
  • Displacement is the shortest distance from A to B.
  • A vector journey is a description of its displacement.
  • The components of a vector describe the journey from A to B e.g. LaTeX: $\left( {\begin{array}{<strong>{20}{c}} x \\ y \end{array}} \right)$ in 2D or LaTeX: $\left( {\begin{array}{</strong>{20}{c}} x \\ y \\ z \end{array}} \right)$ in 3D.
  • Arithmetic can be performed on the components LaTeX: $\left( { + \; - \; \times \; \div } \right)$ e.g.
    LaTeX: $\left( {\begin{array}{<strong>{20}{c}} 1 \\ 2 \end{array}} \right)\,\, + \,\,\left( {\begin{array}{</strong>{20}{c}} 4 \\ 3 \end{array}} \right)\;\; = \;\;\left( {\begin{array}{<strong>{20}{c}} {1\,\, + \,\,4} \\ {2\,\, + \,\,3} \end{array}} \right)\;\; = \;\;\left( {\begin{array}{</strong>{20}{c}} 5 \\ 5 \end{array}} \right)$
  • The magnitude is calculated from the components using the Theorem of Pythagoras e.g. LaTeX: ${\mathbf{u}}\;\; = \;\;\left( {\begin{array}{<strong>{20}{c}} x \\ y \end{array}} \right)$, LaTeX: $\left| {\mathbf{u}} \right|\;\; = \;\;\sqrt {{x^2}\,\, + \,\,{y^2}} $ and LaTeX: ${\mathbf{v}}\;\; = \;\;\left( {\begin{array}{</strong>{20}{c}} x \\ y \\ z \end{array}} \right)$, LaTeX: $\left| {\mathbf{v}} \right|\;\; = \;\;\sqrt {{x^2}\,\, + \,\,{y^2}\,\, + \,\,{z^2}} $

The Zero Vector:

The zero vector has components LaTeX: $\left( {\begin{array}{<strong>{20}{c}} 0 \\ 0 \end{array}} \right)$ in 2D…


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