Problem 661

4. $90^{\circ}$ Hint: According to the parallelogram rule for the sum and difference of vectors, $|\boldsymbol{\alpha}+\boldsymbol{\beta}|,|\boldsymbol{\alpha}-\boldsymbol{\beta}|$ are exactly the lengths of the two diagonals of the parallelogram with $\boldsymbol{\alpha} 、 \boldsymbol{\beta}$ as adjacent sides. Since $|\boldsymbol{\alpha}+\boldsymbol{\beta}|=|\boldsymbol{\alpha}-\boldsymbol{\beta}|$, the parallelogram is a rectangle. Therefore, the angle between $\boldsymbol{\alpha}$ and $\boldsymbol{\beta}$ is $90^{\circ}$.