= Solution
A direct construction, valid for every $0\leq t<1$, is
$$
\boxed{\begin{aligned}
a_1&=(1,0,0),\\
a_2&=(t,\sqrt{1-t^2},0),\\
a_3&=\left(t,\ t\sqrt{\frac{1-t}{1+t}},\
\sqrt{\frac{(1-t)(1+2t)}{1+t}}\right).
\end{aligned}}
$$
The first coordinates give $a_1\cdot a_2=a_1\cdot a_3=t$, and
$$
a_2\cdot a_3=t^2+t(1-t)=t.
$$
The first two norms are one. For the third,
$$
|a_3|^2=t^2+\frac{t^2(1-t)+(1-t)(1+2t)}{1+t}=1.
$$
Thus the required <Gram matrix> is $(1-t)I+t\mathbf1\mathbf1^T$. The column <matrix> is upper triangular and has <determinant>
$$
\boxed{\det(a_1\ a_2\ a_3)=(1-t)\sqrt{1+2t}>0,}
$$
so these <vectors> are <linearly independent>. This is an explicit instance of <unit vectors with a common inner product>; at $t=0$ it reduces to the standard <orthonormal basis>.
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