Why can vitamin K deficiency increase the time needed for blood to clot?

Vitamin K deficiency increases the time required for blood to clot because the vitamin is essential for the synthesis of several key blood-clotting proteins, including prothrombin and factors VII, IX, and X.12

In the liver, vitamin K acts as a necessary cofactor for an enzyme-mediated reaction known as $\gamma$-carboxylation. This process modifies specific glutamic acid residues within these clotting proteins into $\gamma$-carboxyglutamic acid. This unique amino acid is critical because it creates the calcium-binding sites that allow these proteins to interact with cell membranes and function properly.1

When vitamin K is absent, these proteins are synthesized in a deficient, inactive form that cannot bind to calcium or interact with membrane surfaces. Without these functional clotting factors, the coagulation process is impaired, leading to an increase in the time it takes for blood to clot.12