Calculating the Angle Between Two Vectors | A Step-by-Step Guide

Angle between two vectors

The angle between two vectors is a measurement of the separation or rotation between them

The angle between two vectors is a measurement of the separation or rotation between them. It is usually denoted as θ.

To calculate the angle between two vectors, let’s call them vector A and vector B. We can use the dot product formula:

A · B = |A| * |B| * cos(θ)

In this formula, A · B represents the dot product of vector A and vector B, |A| and |B| represent the magnitudes (or lengths) of the vectors, and cos(θ) is the cosine of the angle between them.

If we solve the equation for θ, we get:

θ = cos^(-1)((A · B) / (|A| * |B|))

Here’s a step-by-step guide on how to find the angle between two vectors:

1. Calculate the dot product of the two vectors by multiplying their corresponding components and summing the results.
A · B = A₁ * B₁ + A₂ * B₂ + A₃ * B₃ (for 3D vectors)

2. Find the magnitudes (or lengths) of each vector by taking the square root of the sum of the squares of their components.
|A| = √(A₁^2 + A₂^2 + A₃^2) (for 3D vectors)
|B| = √(B₁^2 + B₂^2 + B₃^2)

3. Substitute the obtained values into the formula:
θ = cos^(-1)((A · B) / (|A| * |B|))

This will give you the measure of the angle in radians. If you want the angle in degrees, you can convert it using the formula: θ_deg = θ_rad * (180/π).

Note: The angle between two vectors is always positive and lies between 0 and 180 degrees (or 0 and π radians) when using the dot product formula.

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