Preheat the oven to 350 degrees. Spray a loaf pan with baking spray and set aside.
In a small bowl, sift together the flour, baking powder, and salt. In another bowl, whisk together the yogurt, 1 cup sugar, the eggs, grapefruit zest, and vanilla. Slowly whisk the dry ingredients into the wet ingredients.
With a rubber spatula, fold the vegetable oil into the batter, making sure it’s all well incorporated. Pour the batter into the prepared pan and bake for about 50 minutes, or until a cake tester inserted in the center of the loaf comes out clean.
Meanwhile, cook the ⅓ cup grapefruit juice and remaining ⅓ cup sugar in a small pan until the sugar dissolves and the mixture is clear. Set aside.
When the cake is done, allow it to cool in the pan for 10 minutes. Carefully place on a baking rack over a sheet pan. While the cake is still warm, pour the grapefruit-sugar mixture over the cake and allow it to soak in. Cool.
For the glaze, combine the powdered sugar and grapefruit juice and drizzle over the cake.