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I came across this notation $u \cdot v = \|u\| \|v\| \cos \theta$ while studying for a linear algebra exam.

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$\|u\|$ is the length of $u$, and $\theta$ is the angle between the vectors $u$ and $v$. (Since you mention the dot product, I’m assuming that $u\cdot v$ isn’t the problem.) – Brian M. Scott Feb 19 '12 at 1:34
There's more than one piece of notation in that equality. – Qiaochu Yuan Feb 19 '12 at 1:34
@BrianM.Scott No the dot product is the not the problem, it was the double pipes surrounding the vectors. – Mike D Feb 19 '12 at 1:41
up vote 4 down vote accepted

The dot product has the formula


The vector norm has the formula


And the angle $\theta$ (or $\Theta$ in your case, I guess) is the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ in Euclidean space. The angle can be visualized directly in two or three dimensions, i.e. $n=2$ or $3$.

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In other words, it is similar to an absolute value. – Mike D Feb 19 '12 at 1:42
@Mike: you should think of it as the length of the vector. This is justified by the Pythagorean theorem. – Qiaochu Yuan Feb 19 '12 at 1:43
@MikeD: Yes. Geometrically, it is the distance from a vector to the origin in Euclidean space. The absolute value is the distance from a number to $0$ on the real line, which can be interpreted as a one-dimensional Euclidean space, so really this is a generalization of the absolute value! (Of course, it isn't the only generalization available...) – anon Feb 19 '12 at 1:45

The equation $$u \cdot v = \| u \| \| v\| \cos \theta$$ means that the dot product between the vectors $u$ and $v$ is equal to the norm of $u$ times the norm of $v$ times cosine of the angle $\theta$ which is the angle between $u$ and $v$.

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