Scalar Product (Dot Product)

BASIC DEFINITION

The scalar product (also known as the dot product) is an algebraic operation between two vectors that returns a real number (a scalar). Unlike the vector product (cross product), which returns a new vector, the scalar product expresses a relationship regarding the direction of two vectors. It is used to calculate the projection of one vector onto another and to determine orthogonality (perpendicularity).

For vectors a = (x₁, y₁, z₁) and b = (x₂, y₂, z₂), the scalar product is defined as:

a · b = x₁·x₂ + y₁·y₂ + z₁·z₂

In a plane (if the z-components are 0), the formula simplifies to:

a · b = x₁·x₂ + y₁·y₂

GEOMETRIC INTERPRETATION

The scalar product can also be expressed using the magnitudes (lengths) of the vectors and the angle between them:

a · b = |a| · |b| · cos(φ),

where φ is the angle between vectors a and b, and 0° ≤ φ ≤ 180° (or 0 ≤ φ ≤ π radians).

From this, it follows:

• If a · b > 0, the angle between the vectors is acute (less than 90°).
• If a · b < 0, the angle is obtuse (greater than 90°).
• If a · b = 0 (and neither vector is a zero vector), the vectors are perpendicular (orthogonal, angle is 90°).

EXAMPLE

Let the given vectors be:
a = (2, 1, –3)
b = (–1, 4, 2)

The scalar product is:
a · b = 2·(–1) + 1·4 + (–3)·2 = –2 + 4 – 6 = –4.

Since the result is negative, the angle between the vectors is obtuse (greater than 90°).

PROPERTIES

• COMMUTATIVITY: a · b = b · a
• DISTRIBUTIVITY OVER VECTOR ADDITION: a · (b + c) = a · b + a · c
• SCALAR MULTIPLICATION: (ka) · b = a · (kb) = k(a · b)
• If at least one vector is the zero vector, then their dot product is 0.
• The scalar product of a vector with a unit vector returns the projection of that vector onto the direction of the unit vector (multiplied by the magnitude of the original vector if it's not a projection of a unit vector). More precisely, the projection of vector a onto vector b is ((a · b) / |b|²) * b.

CONCLUSION

The scalar product is an important operation in vector algebra as it connects the algebraic and geometric representations of vectors. It allows for checking orthogonality, calculating angles and projections, and serves as a foundation in many geometric and physical contexts.