2D & 3D Vectors | Dot & Cross Product | Step-by-Step Solutions
✓ 9 Operations - All vector calculations
✓ 2D & 3D - Both dimensions supported
✓ Step-by-Step - Complete solutions
✓ High Precision - 4 decimal accuracy
2D & 3D Vectors | All Operations | Step-by-Step Solutions | History
Current Operation: Adds corresponding components of two vectors
💡 Tip: Supports both 2D and 3D vectors with complete step-by-step solutions!
x component
y component
z component
x component
y component
z component
No history yet
Addition/Subtraction
Component-wise operations
Dot Product
Scalar result: A·B
Cross Product
Vector perpendicular (3D only)
Magnitude
Length: √(x²+y²+z²)
Vector addition and subtraction with component-wise operations. Calculate resultant vectors for displacement, force, and velocity problems.
Calculate dot product (scalar), cross product (perpendicular vector), and vector magnitude. Essential for physics and engineering calculations.
Angle between vectors, vector projection, normalization, and distance calculations with complete step-by-step working solutions.
A Vector Calculator is specialized mathematical tool performing vector operations calculations essential physics engineering computer graphics mathematics analyzing quantities having both magnitude direction displacement velocity force acceleration represented vectors requiring specific mathematical operations beyond scalar arithmetic vector addition fundamental operation combining two vectors component-wise adding corresponding x y z components resulting resultant vector representing combined effect example adding force vectors finding net force displacement vectors finding total displacement velocity vectors finding resultant velocity parallelogram law vector addition graphically tip-to-tail method algebraically component addition formula A plus B equals Ax plus Bx Ay plus By Az plus Bz essential physics problems multiple forces acting object combining velocities relative motion vector subtraction finding difference between vectors component-wise subtraction corresponding components resulting difference vector representing relative position relative velocity displacement from one point another formula A minus B equals Ax minus Bx Ay minus By Az minus Bz graphically reversing direction B adding negative B essential finding relative quantities change position velocity differences dot product scalar product multiplying vectors producing scalar result formula A dot B equals Ax times Bx plus Ay times By plus Az times Bz also equals magnitude A times magnitude B times cosine theta where theta angle between vectors dot product properties commutative A dot B equals B dot A distributive over addition result scalar not vector dot product zero vectors perpendicular dot product positive acute angle less ninety degrees dot product negative obtuse angle greater ninety degrees essential work calculations physics work equals force dot displacement finding angle between vectors checking perpendicularity projection calculations component analysis cross product vector product creating new vector perpendicular both input vectors three-dimensional only formula A cross B equals determinant i j k Ax Ay Az Bx By Bz resulting Ay Bz minus Az By Az Bx minus Ax Bz Ax By minus Ay Bx cross product properties anti-commutative A cross B equals negative B cross A magnitude equals magnitude A times magnitude B times sine theta direction follows right-hand rule perpendicular both input vectors cross product zero vectors parallel essential torque calculations physics torque equals position cross force angular momentum area parallelogram finding perpendicular vectors computer graphics surface normals making vector calculator indispensable tool modern scientific technical calculations supporting diverse vector operations unified interface.
Our Free Vector Calculator 2025 offers comprehensive advanced features superior functionality supporting both two-dimensional three-dimensional vectors covering all vector operation needs 2D vector mode planar calculations x y components used geometry coordinate plane problems basic physics simple mechanics computer graphics 2D games 3D vector mode spatial calculations x y z components used advanced physics three-dimensional mechanics engineering structural analysis computer graphics 3D modeling game development robotics navigation nine vector operations covering complete vector mathematics vector addition combining vectors component-wise vector subtraction finding differences relative vectors dot product scalar product finding projections angles work calculations cross product vector product perpendicular vectors torque angular momentum 3D only magnitude length distance from origin vector norm angle between vectors calculating angular separation orientation differences normalize unit vector direction preserving magnitude one vector projection component one vector direction another distance between points Euclidean distance spatial separation unique features step-by-step solutions complete working shown every calculation detailed steps component operations intermediate results final answer educational transparent verifiable calculation history automatic tracking last fifteen calculations clickable history items reusing previous results reviewing work verifying computations high precision results four decimal places accuracy maintaining precision complex calculations rounding appropriately scientific engineering standards automatic magnitude calculation result vectors showing magnitude resultant vectors perpendicular vectors projections helping verify results understand vector properties component display clear formatting x y z components parentheses notation standard mathematical convention readable intuitive operation guide contextual information each operation explaining what operation does how used practical applications example calculations error handling robust validation division by zero detection invalid inputs cross-product 2D restriction graceful error messages preventing crashes mobile responsive design flawless operation smartphones tablets desktops touch-friendly inputs readable displays adaptive layouts ensuring accessibility anywhere anytime making most powerful comprehensive user-friendly vector calculator available online completely free unlimited calculations no registration advertisements serving students physics mathematics engineering computer science graphics robotics teachers professors researchers engineers scientists programmers game developers anyone working vectors needing reliable accurate calculation tool updated 2025 standards modern mathematical conventions educational best practices.
Addition: A + B = (Ax + Bx, Ay + By, Az + Bz)
Subtraction: A - B = (Ax - Bx, Ay - By, Az - Bz)
Magnitude: ||A|| = √(Ax² + Ay² + Az²)
Example: (2, 3, 4) + (1, 0, 5) = (3, 3, 9)
Dot Product: A · B = AxBx + AyBy + AzBz
Also: A · B = ||A|| ||B|| cos(θ)
Cross Product: A × B = (AyBz - AzBy, AzBx - AxBz, AxBy - AyBx)
Example: (1, 0, 0) · (0, 1, 0) = 0 (perpendicular)
Angle: θ = arccos((A · B) / (||A|| ||B||))
Projection: proj_B(A) = ((A · B) / (B · B)) × B
Distance: d = ||A - B||
Example: Angle between (1,0) and (1,1) = 45°
Unit Vector: û = A / ||A||
Components: û = (Ax/||A||, Ay/||A||, Az/||A||)
Verification: ||û|| = 1
Example: Normalize (3, 4) = (0.6, 0.8), ||û|| = 1
Component-wise operations: Addition A + B = (Ax+Bx, Ay+By, Az+Bz). Example: (2,3,4) + (1,0,5) = (3,3,9). Subtraction A - B = (Ax-Bx, Ay-By, Az-Bz). Select operation, enter both vectors, click Calculate. Shows step-by-step with component operations. Works 2D and 3D. Perfect for physics force/displacement problems.
Dot product (scalar product): A·B = AxBx + AyBy + AzBz. Also A·B = ||A|| ||B|| cos(θ). Example: (2,3,4)·(1,0,5) = 2+0+20 = 22. Results in scalar. If A·B=0 then perpendicular. Select Dot Product, enter vectors. Calculator shows multiplication steps and sum. Used in work calculations, projections, angle finding.
Cross product creates perpendicular vector (3D only): A×B = (AyBz-AzBy, AzBx-AxBz, AxBy-AyBx). Example: (2,3,4)×(1,0,5) = (15,-6,-3). Direction follows right-hand rule. Magnitude ||A×B|| = ||A|| ||B|| sin(θ). Calculator shows i,j,k component calculations, determinant working. Used in torque, angular momentum, finding perpendicular vectors.
Magnitude (length) using Pythagorean theorem: 2D: ||A|| = √(x²+y²). 3D: ||A|| = √(x²+y²+z²). Example: ||(3,4)|| = √(9+16) = 5. Example 3D: ||(2,3,6)|| = √49 = 7. Calculator shows component squares, sum, square root. Always non-negative. Used for distance, normalizing, calculating speed.
Using dot product: cos(θ) = (A·B)/(||A|| ||B||), then θ = arccos(result). Example: (1,0,0) and (0,1,0) → A·B=0, angle=90° (perpendicular). Calculator shows: dot product, magnitudes, division, arccos in degrees. Range 0° to 180°. Special: 0° parallel same direction, 90° perpendicular, 180° opposite.
Normalization creates unit vector (length 1): û = A/||A||. Example: (3,4,0), ||A||=5 → û=(0.6, 0.8, 0). Verification: ||û||=1. Preserves direction, makes magnitude 1. Calculator shows original, magnitude, division by magnitude, unit vector, verification ||û||=1. Used in direction vectors, graphics normals, physics. Zero vector cannot be normalized.
Projection finds component of A in direction of B: proj_B(A) = ((A·B)/(B·B)) × B. Example: Project (3,4,0) onto (1,0,0) → A·B=3, B·B=1, proj=(3,0,0). Shows how much A points in B direction. Calculator shows: A·B, B·B, scalar multiplier, projection vector. Used in physics (force components), graphics (shadows), decomposing vectors.
Distance is magnitude of difference: d = ||A-B|| = √((Ax-Bx)²+(Ay-By)²+(Az-Bz)²). Example 2D: A=(1,2), B=(4,6) → d=√(9+16)=5. Example 3D: A=(1,2,3), B=(4,6,8) → d=√50≈7.071. Calculator shows: points, difference vector, squares, square root, distance. Euclidean distance formula. Works 2D and 3D.
2D vectors: (x,y), plane geometry, magnitude √(x²+y²), all operations except cross product. 3D vectors: (x,y,z), space geometry, magnitude √(x²+y²+z²), all operations including cross product. Common for both: addition, subtraction, dot product, angle, normalize, projection, distance. Cross product ONLY 3D. Select mode in calculator. 2D simpler, 3D for physics/engineering/graphics.
100% free forever! All features: 2D & 3D calculations, 9 operations (add, subtract, dot, cross, magnitude, angle, normalize, projection, distance), step-by-step solutions, calculation history (15 items), high precision (4 decimals), mobile responsive. No registration, payment, ads, downloads, limits. Works any device/browser. Perfect for students, engineers, scientists, programmers, teachers. Free for everyone!
Our Free Vector Calculator 2025 is the most advanced comprehensive vector mathematics tool supporting both two-dimensional three-dimensional vectors performing all essential vector operations addition subtraction dot product cross product magnitude angle normalization projection distance calculations with complete step-by-step solutions calculation history high precision results completely free unlimited use no registration advertisements serving students engineers scientists programmers game developers teachers anyone working vectors needing reliable accurate calculation tool updated 2025 mathematical standards educational best practices.