Let X = SL3(R)/ SL3(Z), and gt = diag(e 2t , e −t , e −t). Let ν denote the push-forward of the normalized Lebesgue measure on a straight line segment in the expanding horosphere of {gt}t>0 by the map h → h SL3(Z), SL3(R) → X. We give explicit necessary and sufficient Diophantine conditions on the line for equidistribution of each of the following families in X: (1) gt-translates of ν as t → ∞. (2) averages of gt-translates of ν over t ∈ [0, T ] as T → ∞. (3) gt i-translates of ν for some ti → ∞. We apply this dynamical result to show that Lebesgue-almost every point on the planar line y = ax + b is not Dirichlet-improvable if and only if (a, b) / ∈ Q 2 .