Vector Formulas — Operations, Dot Product & Cross Product A complete reference of vector formulas. Covers vector arithmetic, magnitude and unit vectors, dot and cross products, vector projection, the angle between vectors, and a primer on vector calculus (velocity, acceleration, line integrals).
Vector Notation & Basics 2D Vector
v ⃗ = ⟨ v 1 , v 2 ⟩ \vec{v} = \langle v_1,\ v_2 \rangle v = ⟨ v 1 , v 2 ⟩ 3D Vector
v ⃗ = ⟨ v 1 , v 2 , v 3 ⟩ \vec{v} = \langle v_1,\ v_2,\ v_3 \rangle v = ⟨ v 1 , v 2 , v 3 ⟩ Unit Vector Form (3D)
v ⃗ = v 1 i ^ + v 2 j ^ + v 3 k ^ \vec{v} = v_1 \hat{i} + v_2 \hat{j} + v_3 \hat{k} v = v 1 i ^ + v 2 j ^ + v 3 k ^ Vector from Two Points
A B → = ⟨ x B − x A , y B − y A , z B − z A ⟩ \overrightarrow{AB} = \langle x_B - x_A,\ y_B - y_A,\ z_B - z_A \rangle A B = ⟨ x B − x A , y B − y A , z B − z A ⟩ Zero Vector
0 ⃗ = ⟨ 0 , 0 ⟩ \vec{0} = \langle 0, 0 \rangle 0 = ⟨ 0 , 0 ⟩ Magnitude & Unit Vectors Magnitude (2D)
∣ v ⃗ ∣ = v 1 2 + v 2 2 |\vec{v}| = \sqrt{v_1^2 + v_2^2} ∣ v ∣ = v 1 2 + v 2 2 Magnitude (3D)
∣ v ⃗ ∣ = v 1 2 + v 2 2 + v 3 2 |\vec{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2} ∣ v ∣ = v 1 2 + v 2 2 + v 3 2 Unit Vector
v ^ = v ⃗ ∣ v ⃗ ∣ \hat{v} = \frac{\vec{v}}{|\vec{v}|} v ^ = ∣ v ∣ v Direction Angle (2D)
θ = tan − 1 ( v 2 v 1 ) \theta = \tan^{-1}\!\left(\tfrac{v_2}{v_1}\right) θ = tan − 1 ( v 1 v 2 ) Component Form (from magnitude & angle)
v ⃗ = ⟨ ∣ v ⃗ ∣ cos θ , ∣ v ⃗ ∣ sin θ ⟩ \vec{v} = \langle |\vec{v}| \cos\theta,\ |\vec{v}| \sin\theta \rangle v = ⟨ ∣ v ∣ cos θ , ∣ v ∣ sin θ ⟩ Vector Arithmetic Addition
u ⃗ + v ⃗ = ⟨ u 1 + v 1 , u 2 + v 2 ⟩ \vec{u} + \vec{v} = \langle u_1 + v_1,\ u_2 + v_2 \rangle u + v = ⟨ u 1 + v 1 , u 2 + v 2 ⟩ Subtraction
u ⃗ − v ⃗ = ⟨ u 1 − v 1 , u 2 − v 2 ⟩ \vec{u} - \vec{v} = \langle u_1 - v_1,\ u_2 - v_2 \rangle u − v = ⟨ u 1 − v 1 , u 2 − v 2 ⟩ Scalar Multiplication
c v ⃗ = ⟨ c v 1 , c v 2 ⟩ c \vec{v} = \langle c v_1,\ c v_2 \rangle c v = ⟨ c v 1 , c v 2 ⟩ Magnitude of Scaled Vector
∣ c v ⃗ ∣ = ∣ c ∣ ⋅ ∣ v ⃗ ∣ |c \vec{v}| = |c| \cdot |\vec{v}| ∣ c v ∣ = ∣ c ∣ ⋅ ∣ v ∣ Dot Product Dot Product (Component)
u ⃗ ⋅ v ⃗ = u 1 v 1 + u 2 v 2 + u 3 v 3 \vec{u} \cdot \vec{v} = u_1 v_1 + u_2 v_2 + u_3 v_3 u ⋅ v = u 1 v 1 + u 2 v 2 + u 3 v 3 Dot Product (Geometric)
u ⃗ ⋅ v ⃗ = ∣ u ⃗ ∣ ∣ v ⃗ ∣ cos θ \vec{u} \cdot \vec{v} = |\vec{u}| |\vec{v}| \cos\theta u ⋅ v = ∣ u ∣∣ v ∣ cos θ Angle Between Vectors
cos θ = u ⃗ ⋅ v ⃗ ∣ u ⃗ ∣ ∣ v ⃗ ∣ \cos\theta = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| |\vec{v}|} cos θ = ∣ u ∣∣ v ∣ u ⋅ v Orthogonal Test
u ⃗ ⊥ v ⃗ ⟺ u ⃗ ⋅ v ⃗ = 0 \vec{u} \perp \vec{v} \iff \vec{u} \cdot \vec{v} = 0 u ⊥ v ⟺ u ⋅ v = 0 Dot Product with Self
v ⃗ ⋅ v ⃗ = ∣ v ⃗ ∣ 2 \vec{v} \cdot \vec{v} = |\vec{v}|^2 v ⋅ v = ∣ v ∣ 2 Cross Product (3D Only) Cross Product (Determinant)
u ⃗ × v ⃗ = ∣ i ^ j ^ k ^ u 1 u 2 u 3 v 1 v 2 v 3 ∣ \vec{u} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ u_1 & u_2 & u_3 \\ v_1 & v_2 & v_3 \end{vmatrix} u × v = i ^ u 1 v 1 j ^ u 2 v 2 k ^ u 3 v 3 Cross Product (Components)
u ⃗ × v ⃗ = ⟨ u 2 v 3 − u 3 v 2 , u 3 v 1 − u 1 v 3 , u 1 v 2 − u 2 v 1 ⟩ \vec{u} \times \vec{v} = \langle u_2 v_3 - u_3 v_2,\ u_3 v_1 - u_1 v_3,\ u_1 v_2 - u_2 v_1 \rangle u × v = ⟨ u 2 v 3 − u 3 v 2 , u 3 v 1 − u 1 v 3 , u 1 v 2 − u 2 v 1 ⟩ Magnitude of Cross Product
∣ u ⃗ × v ⃗ ∣ = ∣ u ⃗ ∣ ∣ v ⃗ ∣ sin θ |\vec{u} \times \vec{v}| = |\vec{u}| |\vec{v}| \sin\theta ∣ u × v ∣ = ∣ u ∣∣ v ∣ sin θ Anti-Commutative
u ⃗ × v ⃗ = − ( v ⃗ × u ⃗ ) \vec{u} \times \vec{v} = -(\vec{v} \times \vec{u}) u × v = − ( v × u ) Area of Parallelogram
A = ∣ u ⃗ × v ⃗ ∣ A = |\vec{u} \times \vec{v}| A = ∣ u × v ∣ Area of Triangle
A = 1 2 ∣ u ⃗ × v ⃗ ∣ A = \tfrac{1}{2}|\vec{u} \times \vec{v}| A = 2 1 ∣ u × v ∣ Volume of Parallelepiped
V = ∣ u ⃗ ⋅ ( v ⃗ × w ⃗ ) ∣ V = |\vec{u} \cdot (\vec{v} \times \vec{w})| V = ∣ u ⋅ ( v × w ) ∣ Projections Scalar Projection
comp v ⃗ u ⃗ = u ⃗ ⋅ v ⃗ ∣ v ⃗ ∣ \text{comp}_{\vec{v}}\,\vec{u} = \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|} comp v u = ∣ v ∣ u ⋅ v Vector Projection
proj v ⃗ u ⃗ = u ⃗ ⋅ v ⃗ ∣ v ⃗ ∣ 2 v ⃗ \text{proj}_{\vec{v}}\,\vec{u} = \frac{\vec{u} \cdot \vec{v}}{|\vec{v}|^2}\,\vec{v} proj v u = ∣ v ∣ 2 u ⋅ v v Orthogonal Component
u ⃗ ⊥ = u ⃗ − proj v ⃗ u ⃗ \vec{u}_\perp = \vec{u} - \text{proj}_{\vec{v}}\,\vec{u} u ⊥ = u − proj v u Lines & Planes Line (Vector Form)
r ⃗ ( t ) = r ⃗ 0 + t v ⃗ \vec{r}(t) = \vec{r}_0 + t \vec{v} r ( t ) = r 0 + t v Line (Parametric Form)
x = x 0 + a t , y = y 0 + b t , z = z 0 + c t x = x_0 + a t,\ y = y_0 + b t,\ z = z_0 + c t x = x 0 + a t , y = y 0 + b t , z = z 0 + c t Plane (Vector Equation)
n ⃗ ⋅ ( r ⃗ − r ⃗ 0 ) = 0 \vec{n} \cdot (\vec{r} - \vec{r}_0) = 0 n ⋅ ( r − r 0 ) = 0 Plane (Scalar Equation)
a ( x − x 0 ) + b ( y − y 0 ) + c ( z − z 0 ) = 0 a(x - x_0) + b(y - y_0) + c(z - z_0) = 0 a ( x − x 0 ) + b ( y − y 0 ) + c ( z − z 0 ) = 0 Distance Point to Plane
d = ∣ a x 0 + b y 0 + c z 0 + d ∣ a 2 + b 2 + c 2 d = \frac{|a x_0 + b y_0 + c z_0 + d|}{\sqrt{a^2 + b^2 + c^2}} d = a 2 + b 2 + c 2 ∣ a x 0 + b y 0 + c z 0 + d ∣ Vector Calculus Basics Velocity
v ⃗ ( t ) = r ⃗ ′ ( t ) = ⟨ x ′ ( t ) , y ′ ( t ) , z ′ ( t ) ⟩ \vec{v}(t) = \vec{r}\,'(t) = \langle x'(t), y'(t), z'(t) \rangle v ( t ) = r ′ ( t ) = ⟨ x ′ ( t ) , y ′ ( t ) , z ′ ( t )⟩ Speed
∣ v ⃗ ( t ) ∣ = ( x ′ ( t ) ) 2 + ( y ′ ( t ) ) 2 + ( z ′ ( t ) ) 2 |\vec{v}(t)| = \sqrt{(x'(t))^2 + (y'(t))^2 + (z'(t))^2} ∣ v ( t ) ∣ = ( x ′ ( t ) ) 2 + ( y ′ ( t ) ) 2 + ( z ′ ( t ) ) 2 Acceleration
a ⃗ ( t ) = r ⃗ ′ ′ ( t ) \vec{a}(t) = \vec{r}\,''(t) a ( t ) = r ′′ ( t ) Arc Length
L = ∫ a b ∣ r ⃗ ′ ( t ) ∣ d t L = \int_a^b |\vec{r}\,'(t)|\,dt L = ∫ a b ∣ r ′ ( t ) ∣ d t Unit Tangent
T ⃗ ( t ) = r ⃗ ′ ( t ) ∣ r ⃗ ′ ( t ) ∣ \vec{T}(t) = \frac{\vec{r}\,'(t)}{|\vec{r}\,'(t)|} T ( t ) = ∣ r ′ ( t ) ∣ r ′ ( t ) 