# How to find the absolute value of a vector?

In my linear algebra course I keep seeing something like this:

a = {1, 3, 5}

Then in formulas I see this:

|a|

What does this mean, what is the absolute value of a vector? Wouldn't just be {1,3,5}?

Keep in mind that absolute value is distance from zero. So you can use the distance formula to find the absolute value:

$$\sqrt{x^2+y^2+z^2}$$

• It wouldn't happen to be coincidence that this also equals the sqrt of a dot producted with itself would it? Is there any significance there
– Ogen
Mar 11, 2014 at 10:49
• Yep, it is exactly the same thing. For a dot product, you multiply corresponding elements: here, we multiply the x, y, and z values.
– baum
Mar 11, 2014 at 10:59
• Shouldn't that result in the authors question's answer being 5 since the sum of those squares leads to 25 and thus the square root of 25 being 5 and thus the answer too. Also, is this generally accepted as the way to calculate the value of a given vector when given e.g. |v|? Jul 4, 2020 at 13:38
• @user784446 - The sum of squares is $35$, not $25$. May 4, 2022 at 17:48

Absolute value of a vector means taking second norm of the vector i.e. $$\|x\|$$. That means the same thing as $$\sqrt{x_1^2 +x_2^2+...+x_n^2}$$. I don't understand why some top researchers in computer science abuse the notation where $$|x|$$ is widely used for absolute value of scalars in math.

I have also seen papers using $$|x|$$ to mean absolute value of each component of the vector $$x$$. In summary, meaning will depends upon the context and author.

• This is not helpful at all......
– user99914
Jul 2, 2015 at 9:44
• @JohnMa, can you explain me how? or if you know what does $|x|$ exactly mean, where x is a vector?
– CKM
Jul 2, 2015 at 10:00
• If the OP knows what is a norm, probably (s)he won't ask this question. Moreover, you are just changing the notation from $|x|$ to $\|x\|$. How is that helpful? Will the OP get the definition after seeing your answer?
– user99914
Jul 2, 2015 at 10:04
• @JohnMa. Thanks. I should have taken Ogen knowledge level into account.
– CKM
Jul 2, 2015 at 10:15