How do you find the angle between two vectors?

The angle formula and the dot product

The dot product links vector algebra to geometry. For any two non-zero vectors u and v, their dot product equals the product of their lengths multiplied by the cosine of the angle between them: u · v = |u| |v| cos(θ).

To find the angle between two vectors, rearrange this equation to solve for θ: cos(θ) = (u · v) / (|u| |v|). This requires two calculations first: computing the algebraic dot product from components, and finding the length of each vector using the Pythagorean theorem.

Step-by-step calculation in three dimensions

Take vector u = (1, 2, -2) and vector v = (3, 0, 4). First, calculate the dot product by multiplying matching components and adding the results: (1)(3) + (2)(0) + (-2)(4) = 3 + 0 - 8 = -5.

Next, calculate each magnitude. The magnitude of u is √(1² + 2² + (-2)²) = √(1 + 4 + 4) = √9 = 3. The magnitude of v is √(3² + 0² + 4²) = √(9 + 0 + 16) = √25 = 5. The product of their magnitudes is 3 × 5 = 15.

Finally, substitute these values into the formula: cos(θ) = -5 / 15 = -1/3. Use the inverse cosine function on a calculator: arccos(-1/3) ≈ 109.47°. Because the dot product was negative, the angle is obtuse.

Special angles and common pitfalls

Pay close attention to the sign of the dot product. If u · v = 0, cos(θ) = 0, which means the vectors meet at a 90° angle and are perpendicular (orthogonal). If the dot product is positive, the angle is acute. If it is negative, the angle is obtuse.

A common mistake is forgetting parentheses around negative numbers when squaring components, leading to negative values inside the square root. Another error is confusing a vector with a scalar: the dot product yields a single number, not another vector.