I came across this notation $u \cdot v = \|u\| \|v\| \cos \theta$ while studying for a linear algebra exam.
What does this Notation Mean when Calculating the Dot Product
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linear-algebra
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1$\|u\|$ 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.) – 2012-02-19
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1There's more than one piece of notation in that equality. – 2012-02-19
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0@BrianM.Scott No the dot product is the not the problem, it was the double pipes surrounding the vectors. – 2012-02-19
2 Answers
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The dot product has the formula
$$\mathbf{a}\cdot\mathbf{b}=a_1b_1+a_2b_2+\cdots+a_nb_n.$$
The vector norm has the formula
$$\|\mathbf{a}\|=\sqrt{a_1^2+a_2^2+\cdots+a_n^2}.$$
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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0In other words, it is similar to an absolute value. – 2012-02-19
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2@Mike: you should think of it as the _length_ of the vector. This is justified by the Pythagorean theorem. – 2012-02-19
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1@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...) – 2012-02-19
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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 $v$.