Is this quadratic form B(v, v) a symmetric bilinear?

Question

I'm self-studying the Quadrics by following https://people.maths.ox.ac.uk/hitchin/files/LectureNotes/Projective_geometry/Chapter_2_Quadrics.pdf.

On page 25, there is an Example

Consider the quadratic form in R3: $B(v, v) = x_1 x_2 + x_2 x_3 + x_3 x_1$.

The notes also define the quadratic form like this:

Definition 8

A symmetric bilinear form on a vector space $V$ is a map $B : V \times V \rightarrow F$ such that

1. $B(v, w) = B(w, v)$
2. $B(\lambda_1 v_1 + \lambda_2 v_2, w) = \lambda_1 B(v_1, w) + \lambda_2 B(v_2, w)$

My question

But I don't see how this is symmetric. For instance, suppose $v=(v_1, v_2, v_3), w=(w_1, w_2, w_3)$, then $B(v, w)=v_1 w_2 + v_2 w_3 + v_3 w_1$, right? Could someone help me?