Calculate The Angle Between Two Vectors In 2D And 3D Space Using Common Formulas

Angle Between Two Vectors Formula

arccos((u ⋅ v) / (||u|| * ||v||))

The formula to find the angle between two vectors could vary depending on the type of vectors involved (e.g. 2D or 3D vectors, column or row vectors, etc.). Below are the most common formulas:

1. In 2D space:

Let vector `a` = (a1, a2) and vector `b` = (b1, b2). The angle between them can be found using the dot product formula:

θ = cos⁻¹[(a1b1 + a2b2) / (|a| × |b|)]

where:

– θ is the angle between the vectors in radians.
– `|a|` and `|b|` are the magnitudes (lengths) of each vector, defined as:

|a| = sqrt(a1² + a2²)

|b| = sqrt(b1² + b2²)

2. In 3D space:

Let vector `a` = (a1, a2, a3) and vector `b` = (b1, b2, b3). The angle between them can still be found using the dot product, but the formula is slightly different:

θ = cos⁻¹[(a1b1 + a2b2 + a3b3) / (|a| × |b|)]

where:

– θ is the angle between the vectors in radians.
– `|a|` and `|b|` are the magnitudes (lengths) of each vector, defined as:

|a| = sqrt(a1² + a2² + a3²)

|b| = sqrt(b1² + b2² + b3²)

Note that in both 2D and 3D space, the dot product of two perpendicular vectors equals zero, which means their angle is 90 degrees. Conversely, if their dot product equals the product of their magnitudes times -1 (i.e. a · b = -|a| × |b| × cos(θ)), then their angle is 180 degrees (or pi radians).

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