How do I prove that $v=0$, if $v \in \mathbb{R}^n$ is a vector orthogonal to all vectors $x \in \mathbb{R}^n$?

Question

Suppose that $v \in \mathbb{R}^n$ is a vector orthogonal to all vectors $x \in \mathbb{R}^n$. Prove that $v = 0$.

Answer 1

Hint: What is |$v$ $\bullet$ $v$| ?

Answer 2

if v is orthogonal to all of vector of R then for each standarad base we must have $v_i=v*e_i=0$ so v=0

sign * means inner product

Answer 3

If $v$ is orthogonal to all vectors in $\mathbb{R}^n$, then in particular it is orthogonal to itself. What does this, then, imply?

Answer 4

If $v$ is orthogonal to all vectors in $\mathbb R^n$, then as mentioned above $v$ is self orthogonal but then $<v,v>=0$, Now you need to know the definition of inner product (or scalar product in this case).$<v,v>=0$ $\implies v=0$