Isometry/Orthogonal bijective determinant/Exercise

Let ${}V$ be a Euclidean vector space. Let

$\varphi \colon V\longrightarrow V$

denote a linear mapping with the property that the determinant of ${}\varphi$ equals ${}1$ or ${}-1$. Moreover, ${}\varphi$ satisfies the property that orthogonal vectors are mapped to orthogonal vectors. Show that ${}\varphi$ is an isometry.

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