If a linear map sends orthonormal basis on orthonotmal basis then it is an isometry?

Question

Let $(\mathbf{u_1,u_2,u_3})$ and $(\mathbf{v_1,v_2,v_3})$ orthonormal lists. Define $T:\mathbb{R^3}\rightarrow \mathbb{R^3}$ through the lineal extension $T(u_k)=v_k$.

Is T an isometry?

My attempt: I know that the linear map preserves the norm if I use the norm $|t_1\mathbf{u_1}+t_2\mathbf{u_2}+t_3\mathbf{u_3}|=|t_1|+|t_2|+|t_3|$. However, I don't see why distance preserves...In fact, I don't know if this is true...