Dot product of (2,−1) and (5,3)

Given the coordinates of planar vector a are (2, -1), and the coordinates of vector b are (5, 3), find their dot product and the angle between them.

Step-by-step

1. a = (2, -1), b = (5, 3)
   First write both vectors in coordinate form and identify corresponding components.
2. a·b = x₁x₂ + y₁y₂ = 2 × 5 + -1 × 3 = 7
   The dot product in coordinates equals the sum of products of corresponding components; the result is a number, not a vector.
3. |a| = √(2² + -1²) = 2.236068, |b| = 5.830952
   The magnitude of a vector is the principal square root of the sum of squares of its components.
4. cosθ = a·b/(|a||b|) = 0.53687548 ⇒ θ = 57.528809°
   Cosine of the included angle equals the dot product divided by the product of the two magnitudes.
5. a · b = 7 ≠ 0 ⇒ the two vectors are not perpendicular
   A dot product of zero is necessary and sufficient for perpendicularity; it can be used to quickly judge orthogonality.

Why it works

The dot product converts two vectors into a one-dimensional number. Its geometric meaning is the magnitude of one vector multiplied by the projection length of the other onto its direction. In coordinate form, it is the sum of products of corresponding components, and the result is a scalar, not a vector. To find the angle, divide the dot product by the product of their magnitudes and take the arccosine; the denominator is the product of the two magnitudes, not their sum.

Common mistakes

• Writing the dot product as the sum of squares of differences of corresponding components, which is the formula for distance.
• Writing the denominator in the angle formula as the sum of two magnitudes.
• The calculated cosine value exceeds the range of negative one to positive one but arccosine is applied directly, indicating an error earlier.

Related

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