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3-3 (b) Properties of the scalar product

3-3 (b) Properties of the scalar product

Point 2 of 6 — The Scalar (Dot) Product of Two VectorsQuestion bank: 0 questions (0 with pictures)

02 3-3 (b) Properties of the scalar product

List

  • Commutative: A⃗⋅B⃗=B⃗⋅A⃗.
  • Square of a vector: A⃗⋅A⃗=∥A⃗∥2, since the angle is 0∘.
  • Perpendicularity: for non-zero vectors, A⃗⋅B⃗=0 if and only if A⃗⊥B⃗.
  • Distributive: A⃗⋅(B⃗+C⃗)=A⃗⋅B⃗+A⃗⋅C⃗.
  • Scalars: (kA⃗)⋅B⃗=A⃗⋅(kB⃗)=k(A⃗⋅B⃗).
  • Bound: ∣A⃗⋅B⃗∣≤∥A⃗∥ ∥B⃗∥, because ∣cos⁡θ∣≤1.
  • Parallel vectors: A⃗⋅B⃗=±∥A⃗∥∥B⃗∥, with + if they point the same way and − if opposite.
  • The scalar product of unit vectors: i^⋅i^=j^⋅j^=k^⋅k^=1 and i^⋅j^=j^⋅k^=k^⋅i^=0.

Useful identities

  • ∥A⃗+B⃗∥2=∥A⃗∥2+2A⃗⋅B⃗+∥B⃗∥2.
  • (A⃗+B⃗)⋅(A⃗−B⃗)=∥A⃗∥2−∥B⃗∥2.
  • A⃗⋅B⃗=0 also holds if one of the vectors is the zero vector.
✍️ Worked examples

Example 3: ABCD is a square of side 10 cm. Find (a) AB→⋅DC→, (b) AB→⋅BC→, (c) AB→⋅CA→.

Solution:

  • (a) AB→ and DC→ are parallel and in the same direction (θ=0∘): 10×10×1=100.
  • (b) AB→⊥BC→, so the product is 0.
  • (c) CA→=−AC→ and the angle between AB→ and AC→ is 45∘, with AC=102.
  • AB→⋅CA→=−(10×102×cos⁡45∘)=−100.

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