Question

Is the zero vector orthogonal to itself?

Answer 1

No. Even though the dot product is zero, it isn't.

Answer 2

*opinion

Answer 3

(0,0,0,...,0) . (0,0,0...,0) = 0*0 + 0*0 + .. 0*0 = 0

Answer 4

No vector can be orthogonal to itself.

Answer 5

no non-zero vector can be orthogonal to itself

Answer 6

cant be tangent to itself if equal, I say no as well.

Answer 7

Is the zero vector parallel to itself?

Answer 8

Seems more convincing the more I think about it...

Answer 9

(0,0,0...,0) = 1*(0,0,0...,0)

therefore `0` vector is parallel to itself

Answer 10

a vector `u` is parallel to vector `v` iff `u=k*v` , where k is not zero

Answer 11

\[\rm a\cdot b = 0 \implies a\perp b\]
"One thing about math is, you're supposed to follolw the rules" - Gilberat
Strang

Answer 12

no matter how strange the conclusions

Answer 13

maybe k can be zero, but im not sure

Answer 14

I think that the zero vector is orthogonal to itself. All vectors which are orthogonal to itself are called "isotropic vectors"

Answer 15

all vectors? but theres only one vector orthogonal to itself

Answer 16

oops...allvectors which are orthogonal to themselves, are called isotropic vectors

Answer 17

i mean theres only one such vector , the zero vector (im assuming)

Answer 18

I guess that makes it antiparallel to it self also

\[(0,0,...,0)=-(0,0,...,0)\]

Answer 19

So what is the angle between two zero vectors?
\(0\) or \(\pi/2\) , or \(\pi\)?

Answer 20

yes, with respect to the canonical scalar product, the zero vector is orthogonal to itself

Answer 21

since the zero vector has no direction

Answer 22

so the angle is \(\emptyset\)?

Answer 23

yes by definition if you like

Answer 24

Eh, then it makes all possible angles with itself.

Answer 25

@Michele_Laino

I looked up a definition of 'isotropic vector' and this page says the zero vector is not isotropic
http://www.encyclopediaofmath.org/index.php/Isotropic_vector

Answer 26

Please note that the orthogonality is a concept relative to the scalar product which we are considering

Answer 27

yes! Sorry you are right! @UnkleRhaukus Nevertheless the vectors which are orthogonal to themselves exist

Answer 28

I think we need some non-empty angle for the trig functions to work, so it seems that any angle is equally valid

Answer 29

Sorry again @UnkleRhaukus I was too much hurry!!

Answer 30

the angles between zero vectors, depend on which formula you use

Answer 31

better is to say that the scalar product between two vectors depends on which bilinear positive definite form you are using, I think!

Answer 32

or which bilinear form (negative, positive, or undefined) you are using, since not all bilinear forms are positive definite

Answer 33

i thinik we can use two different terms : orthogonal and perpendicular

zero vector is orthogonal to every vector but we cannot say it is perpendicular to any vector

\[ \mathrm{a\cdot b} =0 \iff \mathrm{a} , \mathrm{b} \text{ are orthogonal}\]
\[ \theta =\frac{\pi}{2} \iff\mathrm{a} , \mathrm{b} \text{ are perpendicular}\]

Answer 34

\[a\perp b \iff a\cdot b=0\iff\theta=\frac\pi2\iff a\propto ib\]

Answer 35

what is the difference between perpendicular and orthogonal @ganeshie8 ?

Answer 36

i wouldn't dare saying something like that haha!

Answer 37

angle being 90 degrees can be taken as definition of perpendicularity

Answer 38

https://www.youtube.com/watch?v=5AWob_z74Ks#t=1274

Answer 39

khan is saying there is a difference between the two terms
but in linear algebra we always use them as synonyms

http://mathworld.wolfram.com/OrthogonalVectors.html

Answer 40

http://mathworld.wolfram.com/Perpendicular.html

Answer 41

one thing u need to consider , u should look for axioms systems before work
in vectors they consider 0 vector as the additive identity why ?
1-it has no value
2- it has no direction
in all systems of works the definition of addictive identity is the value that has no defined quantum in our system .

this lead us to this , 0 vector has no direction in first place thus i can't work with it for directions properties .

THE END.

Answer 42

to continue with u , if u considers 0 is orth. to itself that means 0 is orth to all vectors that we can define .
which looks not true to say , however orth means there is no leaner relation or means vectors are independent , but 0 can be written as
0=0.V
which contradict to our claim that its orth.