Pulmonary Embolism (PE): Although the classic presentation of PE includes sudden onset dyspnea, chest pain that worsens with deep breathing (pleuritic chest pain), and possibly cough or hemoptysis, the description given could fit a less typical presentation, especially if the pain radiates to the back and is associated with vertigo due to possible hypoxia or anxiety.

Pneumonia or Pleurisy: Infections or inflammation of the lung tissue or the pleura can cause chest pain that worsens with breathing, including exhaling, and can radiate to the back. Vertigo could be a symptom if the patient is experiencing significant systemic illness or hypoxia.
Panic Attack or Anxiety Disorder: These conditions can cause chest pain, difficulty breathing, and vertigo, especially if the patient is hyperventilating. The pain might worsen with exhaling due to the muscular tension associated with anxiety.
Costochondritis or Tietze's Syndrome: Inflammation of the cartilage that connects the ribs to the breastbone can cause sharp pains in the chest that worsen with deep breathing or movement, potentially radiating to the back. However, vertigo is less commonly associated with these conditions.

Aortic Dissection: A life-threatening condition where the inner layer of the aorta tears, causing severe, tearing chest pain that can radiate to the back. Although the pain typically worsens with inspiration, any chest pain radiating to the back, especially with vertigo (which could indicate decreased blood flow to the brain), necessitates immediate consideration of aortic dissection.
Myocardial Infarction (MI): While MI typically presents with chest pain that worsens with exertion and may radiate to the arm, jaw, or neck, atypical presentations can occur, especially in women, diabetics, or the elderly. Vertigo could be a symptom if the MI leads to arrhythmias or decreased cardiac output.
Pulmonary Hypertension: Can cause chest pain, dyspnea, and potentially vertigo due to decreased cardiac output or hypoxia. The pain might worsen with exertion or deep breathing.

Pneumothorax: Air in the pleural space can cause sudden chest pain that worsens with breathing, potentially radiating to the back. While vertigo is not a common symptom, a large pneumothorax could lead to hypoxia or decreased cardiac output, causing dizziness.
Esophageal Spasm or Rupture: Esophageal conditions can cause chest pain that might worsen with swallowing or breathing and could radiate to the back. Vertigo could occur with severe pain or if the condition leads to hypovolemia or shock.
Mitral Valve Prolapse: A condition where the two valve flaps of the mitral valve do not close smoothly or evenly, which can cause chest pain, palpitations, and potentially vertigo if it leads to arrhythmias or decreased cardiac output. However, this is less likely to cause pain that worsens specifically with exhaling.