Calculating the angle between two vectors involves using their dot product and magnitudes in a specific formula derived from the geometric definition of the dot product.
The most common method to find the angle between two non-zero vectors a and b relies on the formula:
θ = cos⁻¹ [ (a · b) / (|a| |b|) ]
Where:
- θ is the angle between the two vectors.
- a · b represents the dot product of vectors a and b.
- |a| represents the magnitude (or length) of vector a.
- |b| represents the magnitude of vector b.
- cos⁻¹ is the inverse cosine function (also known as arccosine).
To calculate the angle, you need to perform three main steps:
- Calculate the dot product of the two vectors.
- Calculate the magnitude of each vector.
- Plug these values into the formula and find the arccosine of the resulting scalar.
Step 1: Calculate the Dot Product (a · b)
The dot product of two vectors is a scalar quantity. For two vectors in n-dimensional space, say a = <a₁, a₂, ..., a_n> and b = <b₁, b₂, ..., b_n>, the dot product is calculated by summing the products of their corresponding components:
a · b = a₁b₁ + a₂b₂ + ... + a_nb_n
Example:
Let a = <2, 3> and b = <4, -1>.
a · b = (2)(4) + (3)(-1)
a · b = 8 - 3
a · b = 5
Step 2: Calculate the Magnitude of Each Vector (|a| and |b|)
The magnitude of a vector is its length. For a vector v = <v₁, v₂, ..., v_n>, the magnitude is calculated using the Pythagorean theorem in n-dimensions:
|v| = √ (v₁² + v₂² + ... + v_n²)
Example:
Using the same vectors a = <2, 3> and b = <4, -1>:
-
Magnitude of a:
|a| = √ (2² + 3²)
|a| = √ (4 + 9)
|a| = √13 -
Magnitude of b:
|b| = √ (4² + (-1)²)
|b| = √ (16 + 1)
|b| = √17
Step 3: Apply the Angle Formula
Now, substitute the calculated dot product and magnitudes into the formula:
θ = cos⁻¹ [ (a · b) / (|a| |b|) ]
Example:
Using our example values:
- a · b = 5
- |a| = √13
- |b| = √17
First, calculate the denominator:
|a| |b| = (√13)(√17) = √ (13 * 17) = √221
Now, the fraction inside the arccosine:
(a · b) / (|a| |b|) = 5 / √221
Finally, calculate the angle:
θ = cos⁻¹ (5 / √221)
You will need a calculator to find the arccosine of 5 / √221.
5 / √221 ≈ 5 / 14.866 ≈ 0.3363
θ ≈ cos⁻¹ (0.3363) ≈ 70.35 degrees (or ≈ 1.227 radians)
Summary Table
| Step | Description | Formula Example (a=<2,3>, b=<4,-1>) | Result Example |
|---|---|---|---|
| 1. Calculate Dot Product | Sum of products of corresponding components. | a · b = (2)(4) + (3)(-1) | 5 |
| 2. Calculate Magnitudes | Square root of the sum of squared components for each vector. | |a| = √ (2² + 3²), |b| = √ (4² + (-1)²) | √13, √17 |
| 3. Apply Formula | Take the arccosine of the dot product divided by the product of magnitudes. | θ = cos⁻¹ [ 5 / (√13 * √17) ] | ≈ 70.35 degrees |
This method provides a precise way to find the angle between any two non-zero vectors in any number of dimensions.
