Vector Algebra

Two ways to multiply vectors. They give totally different things.

Dot product (scalar product):

a · b = |a| |b| cos θ

= a₁b₁ + a₂b₂ + a₃b₃ in component form. Result is a scalar.

Geometric meaning: projection of a onto b times |b|. Or: how aligned a and b are.

Properties:

• Commutative: a · b = b · a
• Distributive: a · (b + c) = a · b + a · c
• a · a = |a|² (magnitude squared)
• a · b = 0 ⟺ a perpendicular to b (or one is zero vector)
• a · b = |a||b| ⟺ a parallel to b (same direction)

Cross product (vector product):

a × b = |a| |b| sin θ · n̂

where n̂ is perpendicular to both a and b, direction by right-hand rule. Result is a vector.

Component form:
a × b = | i j k ; a₁ a₂ a₃ ; b₁ b₂ b₃ |
= (a₂b₃ − a₃b₂) i − (a₁b₃ − a₃b₁) j + (a₁b₂ − a₂b₁) k

Geometric meaning: |a × b| = area of the parallelogram with sides a and b. Direction perpendicular to the plane containing both.

Properties:

• Anticommutative: a × b = −(b × a)
• Distributive: a × (b + c) = a × b + a × c
• a × a = 0 (always)
• a × b = 0 ⟺ a parallel to b
• |a × b| = |a||b| ⟺ a perpendicular to b

Scalar triple product:

[a b c] = a · (b × c) = | a₁ a₂ a₃ ; b₁ b₂ b₃ ; c₁ c₂ c₃ |

Volume of parallelepiped = |[a b c]|. Three vectors are coplanar ⟺ [a b c] = 0.

Worked example. Find the unit vector perpendicular to both a = i + 2j + 3k and b = 2i − j + k.

a × b = | i j k ; 1 2 3 ; 2 -1 1 | = (2 + 3)i − (1 − 6)j + (−1 − 4)k = 5i + 5j − 5k.

|a × b| = √(75) = 5√3. Unit vector = (5i + 5j − 5k) / (5√3) = (i + j − k) / √3.