Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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

share|improve this question
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.) –  Brian M. Scott Feb 19 '12 at 1:34
1  
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

2 Answers 2

up vote 3 down vote accepted

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 $3$.

share|improve this answer
    
In other words, it is similar to an absolute value. –  Mike D Feb 19 '12 at 1:42
2  
@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
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...) –  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 $v$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.