Vector Identities | Physics Library

It is difficult to get anywhere in physics without a firm understanding of vectors and their common operations. Here, we will give vector identities as a reference. Basic terminology to keep straight.




Operation	Symbol



Gradient	∇f
Laplacian	∇²
divergence	∇⋅
curl	∇×

Vector Magnitude

A = |A | = ∘ -----------------
2 2 2
Ax + Ay + Az
A = √ ------
A ⋅ A

scalar product (Dot Product)

A ⋅ B = A_xB_x + A_yB_y + A_zB_z
A ⋅ B = |A | |B| cos 𝜃

vector product (cross product)

A × B = (AyBz − AzBy ) î + (AzBx − AxBz ) ĵ + (AxBy − AyBx ) k

It can be easier to remember with determinant formulation

A × B = | |
|| ˆi ˆj ˆk ||
||Ax Ay Az ||
|B B B |
x y z = (AyBz − AzBy ) î + (AzBx − AxBz ) ĵ + (AxBy − AyBx ) k

Vector Triple Product, aka. BAC CAB

A × (B × C ) = B (A ⋅ C ) − C (A ⋅ B )

scalar triple product

A ⋅ (B × C ) = B ⋅ (C × A ) = C ⋅ (A × B )

Gradient

∇f = ∂∂fx î + ∂∂fy ĵ + ∂∂fz k

Gradient identities

∇ (f + g) = ∇f + ∇g
∇ (αf ) = α∇f
∇ (f g) = f∇g + g∇f
∇ (f ∕g) = (g∇f−f∇g)
----g2---

Divergence

∇⋅ A = ∂Ax-
∂x + ∂Ay-
∂y + ∂Az
∂z

Divergence of the cross product

∇⋅ (A × B ) = B ⋅ (∇ × A ) − A ⋅ (∇ × B )

Divergence of the curl

∇⋅ (∇ × A ) = 0

Laplacian Identities

∇× (∇ × A ) = ∇ (∇ ⋅ A ) −∇²A

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