A Primer for Cross Product Calculations

October 29, 2025

🧭 A Primer for Cross Product Calculations

Right Hand Rule for Cross Product. Source: https://commons.wikimedia.org/wiki/File:Right-hand_rule_for_cross_product.png

1. Basis Vectors

î = (1, 0, 0), ĵ = (0, 1, 0), k̂ = (0, 0, 1)

These are unit and mutually perpendicular vectors.

2. Dot Product (for reference)

𝐀 · 𝐁 = |𝐀| |𝐁| cos θ

î · ĵ = ĵ · k̂ = k̂ · î = 0
î² = ĵ² = k̂² = 1

3. Definition of the Cross Product

𝐀 × 𝐁 = |𝐀| |𝐁| sin θ n̂

θ is the angle from 𝐀 to 𝐁.
n̂ is a unit vector perpendicular to both 𝐀 and 𝐁.
The direction of n̂ follows the right-hand rule (anticlockwise = positive, clockwise = negative).

4. Fundamental Basis Cross Products

î × ĵ =  k̂
ĵ × k̂ =  î
k̂ × î =  ĵ

Reversing the order changes the sign:

ĵ × î = −k̂
k̂ × ĵ = −î
î × k̂ = −ĵ

And any vector crossed with itself is zero:

î × î = ĵ × ĵ = k̂ × k̂ = 0

5. Expansion in Component Form

𝐀 = a₁ î + a₂ ĵ + a₃ k̂
𝐁 = b₁ î + b₂ ĵ + b₃ k̂

By distributivity:

𝐀 × 𝐁 =
(a₂b₃ − a₃b₂) î +
(a₃b₁ − a₁b₃) ĵ +
(a₁b₂ − a₂b₁) k̂

6. Determinant Representation

      |  î   ĵ   k̂  |
𝐀 × 𝐁 = | a₁  a₂  a₃ |
         | b₁  b₂  b₃ |

Expanding this determinant:

𝐀 × 𝐁 =
 î |a₂ a₃|
    |b₂ b₃| 
− ĵ |a₁ a₃|
     |b₁ b₃|
+ k̂ |a₁ a₂|
     |b₁ b₂|

This is simply a compact way to remember the component formula above.

7. Computational Formula

For quick calculations:

𝐀 × 𝐁 = (a₂b₃ − a₃b₂,  a₃b₁ − a₁b₃,  a₁b₂ − a₂b₁)

8. Geometric Interpretation

𝐀 × 𝐁 is perpendicular to both 𝐀 and 𝐁.
Its magnitude equals the area of the parallelogram spanned by 𝐀 and 𝐁:

|𝐀 × 𝐁| = |𝐀| |𝐁| sin θ

The direction follows the right-hand rule.

✅ Summary Table

Operation	Result
î × ĵ	+ k̂
ĵ × k̂	+ î
k̂ × î	+ ĵ
Reversed order	Negative result
Same vectors	0
Magnitude	\|𝐀\| \|𝐁\| sin θ
Direction	Perpendicular to both (right-hand rule)

From this foundation, all 3D cross products can be computed directly and visualised geometrically.

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