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<title>Course on Practical Neuroimaging in Python — Practical neuroimaging analysis</title>
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<div class="section" id="the-dot-product-of-two-unit-vectors-in-2d">
<h1>The dot product of two unit vectors in 2D<a class="headerlink" href="#the-dot-product-of-two-unit-vectors-in-2d" title="Permalink to this headline">¶</a></h1>
<p>The dot product of two unit vectors <span class="math">\(\hat{a}, \hat{b}\)</span> is <span class="math">\(\cos \theta\)</span>, where
<span class="math">\(\theta\)</span> is the angle between the unit vectors.</p>
<div class="section" id="proof">
<h2>Proof<a class="headerlink" href="#proof" title="Permalink to this headline">¶</a></h2>
<p>We remember the rotation formula from <a class="reference internal" href="rotation_2d.html"><span class="doc">Formula for rotating a vector in 2D</span></a>:</p>
<div class="math">
\[\begin{split}x_2 = \cos \beta x_1 - \sin \beta y_1 \\
y_2 = \sin \beta x_1 + \cos \beta y_1\end{split}\]</div>
<p>Now consider the unit vector <span class="math">\(\hat{a} = (x_1, y_1)\)</span> rotated by <span class="math">\(\theta\)</span> to get
<span class="math">\(\hat{b} = (x_2, y_2)\)</span>:</p>
<div class="math">
\[\begin{split}\hat{a} \cdot \hat{b} = x_1 x_2 + y_1 y_2 \\
= x_1 (\cos \theta x_1 - \sin \theta y_1) + y_1 (\sin \theta x_1 + \cos
\theta y_1) \\
= \cos \theta x_1^2 - \sin \theta x_1 y_1 + \sin \theta x_1 y_1 + \cos
\theta y_1^2 \\
= \cos \theta (x_1^2 + y_1^2)\end{split}\]</div>
<p>Because we have assumed <span class="math">\(\hat{a}\)</span> is a unit vector, <span class="math">\(\| \hat{a} |\ = x_1^2 + y_1^2 = 1\)</span> and:</p>
<div class="math">
\[\hat{a} \cdot \hat{b} = \cos \theta\]</div>
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<div class="section" id="dot-product-of-not-unit-vectors">
<h2>Dot product of not-unit vectors<a class="headerlink" href="#dot-product-of-not-unit-vectors" title="Permalink to this headline">¶</a></h2>
<p>Now take two non-unit vectors <span class="math">\(\vec{x}, \vec{y}\)</span>:</p>
<div class="math">
\[\begin{split}\hat{x} = \frac{1}{\| \vec{x} \|}\vec{x} \\
\hat{y} = \frac{1}{\| \vec{y} \|}\vec{y}\end{split}\]</div>
<p>From the rules of the dot product and the result from the unit vectors above:</p>
<div class="math">
\[\vec{x} \cdot \vec{y} = \| \vec{x} \| \| \vec{y} \| \cos \theta\]</div>
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<li><a class="reference internal" href="#dot-product-of-not-unit-vectors">Dot product of not-unit vectors</a></li>
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