Introduction
This free vector calculator lets you solve vector math problems step by step. You can add, subtract, find the dot product, cross product, magnitude, unit vector, angle between vectors, and more, all in 2D or 3D. Just pick an operation, type in your numbers, and hit Calculate. The tool shows you every step so you can learn how the math works, not just get the answer.
This calculator helps you check your work and build your skills in linear algebra, physics, and precalculus. You can also enter a full expression like dot (1,2,3) (4,-5,6) for a quick solve. A built-in "Check My Answer" feature lets you test your own result against the correct one. Try one of the ready-made examples below to get started right away.
How to Use Our Vector Calculator
Enter your vectors and pick an operation. The calculator will give you the answer, a step-by-step solution, a visual diagram, and a plain-English explanation of what the result means.
Pick an operation. Click one of the 12 operation cards, like Dot Product, Cross Product, Magnitude, or Add. The formula bar below the cards will update to show you the math formula being used.
Choose your input mode. Use Structured mode to type each vector component into its own box. Use Expression mode to type a full math expression, like dot (1,2,3) (4,-5,6), into a single text field.
Set the dimension. Click 2D if your vectors only have x and y parts. Click 3D if they also have a z part. Some operations, like Cross Product, only work in 3D and will lock this setting for you.
Enter Vector u. Type a number into each component box (x, y, and z) for your first vector. This vector is used in every operation.
Enter Vector v. For operations that need two vectors, like Dot Product or Angle Between, type the x, y, and z values for your second vector.
Enter Vector w. This third vector only appears for the Scalar Triple Product, which finds the volume formed by three vectors.
Enter Scalar k. This number field only appears for Scalar Multiplication. Type the number you want to scale your vector by.
Enter Magnitude and Angle. These fields only appear when you convert from polar or spherical form back to components. Type the length and angle(s) in degrees.
Click Calculate. Press the blue Calculate button to see your results. You can also just press Enter on your keyboard from any input box.
Read your results. The output includes your formatted input, each solution step, the final answer, an interpretation in plain words, a component breakdown, and a vector diagram. Use the Display Options checkboxes to show or hide any of these sections.
Check your own work. Scroll to the Check My Answer section at the bottom. Type in what you think the answer is, then click Check to see if it matches the computed result.
What Are Vectors?
A vector is a math object that has both a size (called magnitude) and a direction. Think of it like an arrow. The arrow's length tells you how big it is, and the way it points tells you its direction. Vectors are written as a list of numbers inside angle brackets, like ⟨3, 4⟩ in 2D or ⟨1, 2, 5⟩ in 3D. Each number is called a component. The components tell you how far the vector reaches along each axis: x, y, and sometimes z.
Why Do Vectors Matter?
Vectors show up everywhere in science and everyday life. When you push a box across the floor, the force you use has a strength and a direction. That's a vector. Wind speed, the path a car drives, and even the pull of gravity are all described using vectors. In math, vectors are a key part of linear algebra, a subject used in physics, engineering, computer graphics, and data science.
Common Vector Operations
Vector addition combines two vectors into one resultant vector by adding their matching components. Subtraction finds the difference between two vectors. The dot product multiplies two vectors together and returns a single number (a scalar). It tells you how much two vectors point in the same direction. The cross product takes two 3D vectors and gives back a new vector that is perpendicular to both of them.
The magnitude of a vector is its length, found using the Pythagorean theorem. A unit vector keeps the same direction but shrinks the length to exactly 1. Scalar multiplication stretches or shrinks a vector by a number. Vector projection finds the part of one vector that falls along another, like a shadow. The scalar triple product uses three vectors to find the volume of a 3D box shape called a parallelepiped. If you need to work with the angle between vectors, the formula relies on the inverse cosine.
2D vs. 3D Vectors
A 2D vector lives on a flat plane and has two components: x and y. A 3D vector adds a third component, z, and works in space. Most operations work in both 2D and 3D. The cross product and scalar triple product only work with 3D vectors because they rely on a third dimension to define a perpendicular direction or a volume. Vectors also play a central role in physics problems involving torque, momentum, projectile motion, and displacement.