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Vectors & Vector Operations

COMP116 Lectures

$n$-vectors

  • A set of numbers, $H$.

    • $H$ is one of: $\Bbb{N,W,Z,Q;R}$.
  • A natural number, $n\in\Bbb{N}$.
  • An $n$-vector over $H$, has $n$ components: \(\underline x = \langle x_1,x_2,\ldots,x_n\rangle\)
  • Each $x_k\in H$.

This is not a set: we cam have $x_s=x_t$ and $s\neq t$.

  • The order of components matters: $\langle 1,2,1\rangle\neq\langle1,1,2\rangle$.

Notation & Operands

In the presentation the lecturer will use the following notation:

  • Arbitrary vectors: $\mathbf{\underline x,\underline y, \underline u, \underline v; \underline w}$.
  • Writing vectors in full: $\langle u_1,u_2,\ldots,u_m\rangle$.
  • For points we use: $(u_1,u_2,\ldots,u_m)$.

As this notation for arbitrary vectors is tedious to type the following will be used:

  • Arbitrary vectors: $\vec x, \vec y,\vec u,\vec v, \vec w$.

Operations

  • Addition: $+$
  • Scalar multiple: $*$
  • Size or norm: $\Vert u\Vert$ or $\Vert u\Vert_2$
  • Product: $\cdot$ or $\times$
    • You can use: $u\cdot v=\Vert u\Vert \Vert v\Vert cos\theta$.

Given $n$-vectors: $\vec u,\vec v,\vec w$ etc. and a constant $\alpha\in H$.

  • Addition
    • $\vec w=\vec u+\vec v$
    • $w_k+v_k\forall k1\leq k\leq n$ This is to say that if you want to add two vectors they need to have the same number of components.

  • Scalar Multiple

    • $\vec w=\alpha\vec u$

    • $w_k=\alpha u_k\forall k1\leq k\leq n$

      This means that you just times all the components by the scalar.

  • Norm

    • $\Vert u\Vert=\sqrt{\sum^n_{k=1}\vert u_k\vert^2}$

      This is just the Pythagorean theorem.

  • Product

    • Definition is not given.

Attributes

  • A vector $\vec u$ has a size $\Vert u\Vert$ and a direction.
  • A vector doesn’t not have a position.

  • The operation can be interpreted geometrically.

  • The length of a vector resulting from adding two vectors may be written in terms of basic trigonometric relations.

Dot Product

For two vectors, their dot product is the cosine of the angle between them. This gives the following cases:

  • Right angles give a dot product of 0.
  • Parallel vectors give a dot product of -1.
    • Position doesn’t matter and thus the direction doesn’t change this value.

Special Vectors

  • The zero vector $\vec 0$:
    • All values are zero.
  • The basis vector $e_k$:
    • All components are zero except the $k^{th}$ which is equal to 1.
    • There are $n$ basis vectors, one for each $k$.
  • Unit vectors:
    • Those whose magnitude is 1.
  • Orthogonal vectors:
    • At right angles, whose dot product is 0.