Dot Product

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The dot product of two vectors yields a scalar, which is computed as follows:

\[(x_1, y_1, z_1) \cdot (x_2, y_2, z_2) = x_1 x_2 + y_1 y_2 + z_1 z_2\]

Without meaning this operator wouldn’t have much use. So, what does this scalar represent? What can we use it for?

To answer this question, you need to know of an alternative way or computing the dot product:

\[\vec u \cdot \vec v = |\vec u| \cdot |\vec v| \cdot cos(\theta)\]

In other words, the dot product equals the multiplication of the length of both vectors and the cosine of the angle between them.

This alternative formula makes it possible to use the dot product to compute the length of vectors and the angle between them.

1. Determining the Length

To determine the length of a vector \(\vec{v}(x,y,z)\) we can multiply \(\vec v\) by itself. In this case the dot product yields \(|\vec v| \cdot |\vec v| \cdot \cos(0^\circ)\). Because \(\cos(0^\circ) = 1\), we ultimately get

\[ x^2 + y^2 + z^2 = |\vec v|^2\]

Taking the square root yields the length of the vector.

2. Determining the Angle

Finding the angle between the vectors is a bit more complicated. We know

\[\vec u \cdot \vec v = |\vec u| \cdot |\vec v| \cdot cos(\theta)\]

and we are interested in \(\cos(\theta)\). We move the vector lengths to the left hand side of the equation. Computing the length of both vectors is a problem already solved, so we get

\[\cos(\theta) = \frac{\vec u \cdot \vec v}{|\vec u| \cdot |\vec v|} = \frac{\vec u \cdot \vec v}{\sqrt{\vec u \cdot \vec u} \cdot \sqrt{\vec v \cdot \vec v}}\]

So, now we have \(\cos(\theta)\) from which it is possible to compute \(\theta\). However, this extra step will seldom be necessary: having \(\cos(\theta)\) itself will often be exactly what we need in this course.

Lastly, the dot product allows us to easily find out whether two vectors are perpendicular to each other. If they are, their dot product is 0. This is due to the fact that \(\cos(90^\circ) = 0\).

\[\vec u \perp \vec v \qquad \Leftrightarrow \qquad \vec u \cdot \vec v = 0\]