Scalar Product or Dot Product

Subtopic 1 — Scalar Product or Dot Product

Scalar Product (Dot Product)

Terms Used in the Formula

In: A⃗ · B⃗ = AB cos θ A is the magnitude of A⃗. B is the magnitude of B⃗. θ is the smaller angle between the vectors. The dot (·) represents scalar multiplication. The answer is a scalar quantity.

Physical Meaning of Dot Product

The dot product measures the component of one vector acting along the direction of another vector. The component of B⃗ along A⃗ is: B cos θ Therefore: A⃗ · B⃗ = A(B cos θ) Similarly: A⃗ · B⃗ = B(A cos θ) A larger dot product indicates that the vectors act more strongly in the same direction.

Dot Product at Special Angles

Parallel Vectors When θ = 0°: A⃗ · B⃗ = AB cos 0° = AB The dot product has its maximum positive value. Perpendicular Vectors When θ = 90°: A⃗ · B⃗ = AB cos 90° = 0 Therefore, the dot product of two perpendicular vectors is zero. Opposite Vectors When θ = 180°: A⃗ · B⃗ = AB cos 180° = −AB The dot product has its maximum negative value.

Dot Product Using Components

Consider: A⃗ = Aₓ î + Aᵧ ĵ + Az k̂ and: B⃗ = Bₓ î + Bᵧ ĵ + Bz k̂ Their dot product is: A⃗ · B⃗ = AₓBₓ + AᵧBᵧ + AzBz For two-dimensional vectors: A⃗ · B⃗ = AₓBₓ + AᵧBᵧ

Dot Products of Unit Vectors

A unit vector has a magnitude of one. Therefore: î · î = 1 ĵ · ĵ = 1 k̂ · k̂ = 1 Different Cartesian unit vectors are perpendicular. Therefore: î · ĵ = 0 ĵ · k̂ = 0 k̂ · î = 0

Properties of Dot Product

Commutative Property A⃗ · B⃗ = B⃗ · A⃗ Distributive Property A⃗ · (B⃗ + C⃗) = A⃗ · B⃗ + A⃗ · C⃗ Multiplication of a Vector by Itself A⃗ · A⃗ = A² Therefore: A = √(A⃗ · A⃗) Scalar Multiplication (mA⃗) · (nB⃗) = mn(A⃗ · B⃗)

Angle Between Two Vectors

From: A⃗ · B⃗ = AB cos θ We get: cos θ = (A⃗ · B⃗) / AB Therefore: θ = cos⁻¹((A⃗ · B⃗) / AB) If the dot product is: Positive, the angle is less than 90°. Zero, the vectors are perpendicular. Negative, the angle is greater than 90°.

Applications of Dot Product

The dot product is used to calculate

• Work Done: W = F⃗ · s⃗ = Fs cos θ
• Instantaneous Power: P = F⃗ · v⃗
• Component of One Vector Along Another: A∥ = A cos θ
• Angle Between Two Vectors: θ = cos⁻¹((A⃗ · B⃗) / AB)

Solved Examples — Dot Product