Solution
= Solution
Expanding a sum of squares gives the $n$-dimensional <Lagrange identity for the cross product>:
$$
\frac12\sum_{i,j}(x_iy_j-x_jy_i)^2
=\sum_{i,j}x_i^2y_j^2-\sum_{i,j}x_ix_jy_iy_j
=|x|^2|y|^2-(x\cdot y)^2.
$$
The left-hand side is nonnegative, so
$$
\boxed{|x|^2|y|^2\geq|x\cdot y|^2}.
$$