Dot Product

Geometric Definition

Let $\vec v \cdot \vec w$ denote the dot product between vectors $\vec v$ and $\vec w$

  • definition: $\vec v \cdot \vec w = \| \vec v \| \cdot \| \vec w \| \cdot \cos \theta$ where $\theta$ is the angle between $\vec v$ and $\vec w$
  • $\| \vec v \|$ denotes the length of $\vec v$
  • if two vectors are perpendicular, then $\cos \theta = 0$ and thus $\vec v \cdot \vec w = 0$
  • if they co-directional, then $\theta = 0$ and $\vec v \cdot \vec w = \| \vec v \| \cdot \| \vec w \|$
  • consequently, we have $\vec v \cdot \vec v = \| \vec v \|^2$

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Projections

Dot product is a projection:

  • let's project $\vec v$ onto $\vec w$: $\text{proj}_{\vec w} (\vec v) = \| \vec v \| \cos \theta$ (by the $\cos$ definition)
  • we're interested only in the direction of $\vec w$, so let's normalize it to get $\hat w = \vec w / \| \vec w \|$ - it's the unit vector in the direction $\vec w$
  • $\vec v \cdot \hat w$ - dot product of $\vec v$ and some unit vector
    • $\vec v \cdot \hat w = \| \hat w \| \| \vec v \| \cos \theta = \| \vec v \| \cos \theta$
    • it is a projection in the direction of $\vec w$
  • so $\vec v \cdot \hat w$ corresponds to the projection of $\vec v$ onto $\vec w$
  • $\text{proj}_{\vec w} (\vec v) = \vec v \cdot \hat w = \| \vec v \| \cos \theta$ - is the length of this projection
  • thus, $\vec v \cdot \vec w = \| \vec v \| \| \vec w \| \cos \theta = \| \vec w \| \cdot \text{proj}_{\vec w} (\vec v) = \| \vec v \| \cdot \text{proj}_{\vec v} (\vec w)$


$p = \text{proj}_{\vec u} (\vec v) = \vec v \cdot \hat u = \| \vec v \| \cos \theta$

  • it can be positive or negative
  • positive when angle between vectors is less than $90^o$
  • negative when angle between vectors is greater than $90^o$

svm-vectors-projection.png


The Cosine Theorem

Why does this geometric definition make sense?

  • consider vectors $\vec u$, $\vec v$ and $\vec w$
  • let $\vec v + \vec u = \vec w$ or $\vec u = \vec w - \vec v$
  • d9f8b10734864b92bdcf9cf5ac92a0dc.png
  • $\| \vec u \|^2 = \| \vec w - \vec v \|^2 = (\vec w - \vec v) \cdot (\vec w - \vec v) = \| \vec w \|^2 + \| \vec v \|^2 - 2 \cdot \vec w \vec v$
  • by the Cosine Theorem we know that
    • $\| \vec u \|^2 = \| \vec w - \vec v \|^2 = \| \vec w \|^2 + \| \vec v \|^2 - 2 \cdot \| \vec w \| \cdot \| \vec v \| \cdot \cos \theta$
    • so $\| \vec w \| \cdot \| \vec v \| \cdot \cos \theta = \cfrac{1}{2} (\| \vec w \|^2 + \| \vec v \|^2 - \| \vec w - \vec v \|^2) = \cfrac{1}{2} (\| \vec w \|^2 + \| \vec v \|^2 - \| \vec w \|^2 - \| \vec v \|^2 + 2 \cdot \vec w \vec v) = \vec w \cdot \vec v$
  • thus $\vec w \cdot \vec v = \| \vec w \| \cdot \| \vec v \| \cdot \cos \theta$
  • i.e. the definition makes sense from the The Cosine Theorem point of view


Algebraic Definition

For two vectors $\mathbf v, \mathbf w \in \mathbb R^n$ we define the dot product as $\mathbf v^T \mathbf w = \sum\limits_{i = 1}^n v_i w_i$


Vector Orthogonality

$\mathbf v \; \bot \; \mathbf w \iff \mathbf v^T \mathbf w = 0$

Why?

  • by the Pythagoras theorem: $\| \mathbf v \|^2 + \| \mathbf w \|^2 = \| \mathbf v + \mathbf w \|^2$
  • $\mathbf v^T \mathbf v + \mathbf w^T \mathbf w = (\mathbf v + \mathbf w)^T (\mathbf v + \mathbf w) = \mathbf v^T \mathbf v + \mathbf v^T \mathbf w + \mathbf w^T \mathbf v + \mathbf w^T \mathbf w$
  • or $\mathbf v^T \mathbf w + \mathbf w^T \mathbf v = 0$
  • note that $\mathbf v^T \mathbf w = \mathbf w^T \mathbf v$, so we have
  • $2 \mathbf v^T \mathbf w = 0$ or $\mathbf v^T \mathbf w = 0$


Equivalence of Definitions

Both definitions are equivalent.

I.e. $\vec v \cdot \vec w = \mathbf v^T \mathbf w = \| \vec v \| \| \vec w \| \cos \theta = \sum\limits_{i = 1}^n v_i w_i$

Why?

  • let $\vec e_1, \ ... \ , \vec e_n$ be the standard basis in $\mathbb R^n$
  • these vectors are orthonormal, i.e. $\vec e_i \cdot \vec e_j = \begin{cases}

1 & \text{if } i = j \\ 0 & \text{if } i \ne j \\ \end{cases}$

  • let's consider a dot product $\vec v \cdot \vec e_i$: $\vec v \cdot \vec e_i = \| \vec v \| \cos \theta = v_i$ - it's a projection of $\vec v$ onto $ \vec e_i$ - which is $i$th component of $\vec v$
  • we can project $\vec v$ and $\vec w$ onto the entire basis and get
  • $\vec v = \sum\limits_{i = 1}^n v_i \cdot \vec e_i$ and $\vec w = \sum\limits_{i = 1}^n w_i \cdot \vec e_i$
  • So, $\vec v \cdot \vec w = \vec v \cdot \sum\limits_{i = 1}^n w_i \vec e_i = \sum\limits_{i = 1}^n w_i (\vec v \cdot \vec e_i) = \sum\limits_{i = 1}^n v_i w_i$

$\square$


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