Vector Inputs
î (1,0)
ĵ (0,1)
Diagonal
Perpendicular
Operations
A + B
A − B
Scalar kA
Dot A·B
Magnitude
Unit Vector
Angle Between
Projection
1.0
Properties
Interactive Demos
Parallelogram Rule
Visualize addition
Dot Product = 0
Perpendicular vectors
Projection
See projection change
Basis Vectors
Decompose into components
Step-by-Step Solution
Select an operation and click Compute to see the solution.
Quick Reference
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Vector Operations
$\vec{A}+\vec{B} = (a_x+b_x,\; a_y+b_y,\; a_z+b_z)$
$\vec{A}-\vec{B} = (a_x-b_x,\; a_y-b_y,\; a_z-b_z)$
$k\vec{A} = (k\,a_x,\; k\,a_y,\; k\,a_z)$
$|\vec{A}| = \sqrt{a_x^2 + a_y^2 + a_z^2}$
$\hat{A} = \dfrac{\vec{A}}{|\vec{A}|}$
Dot Product
$\vec{A}\cdot\vec{B} = a_x b_x + a_y b_y + a_z b_z$
$\vec{A}\cdot\vec{B} = |\vec{A}|\,|\vec{B}|\cos\theta$
Commutative: $\vec{A}\cdot\vec{B}=\vec{B}\cdot\vec{A}$
Distributive: $\vec{A}\cdot(\vec{B}+\vec{C})=\vec{A}\cdot\vec{B}+\vec{A}\cdot\vec{C}$
Cross Product (3D)
$\vec{A}\times\vec{B} = \begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\a_x&a_y&a_z\\b_x&b_y&b_z\end{vmatrix}$
$= (a_y b_z - a_z b_y)\hat{i} - (a_x b_z - a_z b_x)\hat{j} + (a_x b_y - a_y b_x)\hat{k}$
Anti-commutative: $\vec{A}\times\vec{B}=-(\vec{B}\times\vec{A})$
$|\vec{A}\times\vec{B}| = |\vec{A}|\,|\vec{B}|\sin\theta$ (parallelogram area)
Projection
$\text{proj}_{\vec{B}}\vec{A} = \dfrac{\vec{A}\cdot\vec{B}}{|\vec{B}|^2}\,\vec{B}$
$\text{comp}_{\vec{B}}\vec{A} = \dfrac{\vec{A}\cdot\vec{B}}{|\vec{B}|}$ (scalar projection)