Find vector components from magnitude and angle. Build vectors from points and verify direction fast. Clean steps make component calculations easier for every learner.
| Case | Input | Component Form | Magnitude | Direction |
|---|---|---|---|---|
| 2D magnitude and angle | r = 10, θ = 30° | <8.6603, 5> | 10 | 30° |
| Two 2D points | A(2, 1), B(7, 5) | <5, 4> | 6.4031 | 38.6598° |
| Existing components | x = -3, y = 4 | <-3, 4> | 5 | 126.8699° |
| Two 3D points | A(1, 2, 3), B(5, 7, 9) | <4, 5, 6> | 8.775 | 3D vector |
x = r cos(θ)
y = r sin(θ)
Vector = <x, y>
x = x₂ - x₁
y = y₂ - y₁
Vector = <x₂ - x₁, y₂ - y₁>
|v| = √(x² + y²)
θ = atan2(y, x)
x = x₂ - x₁
y = y₂ - y₁
z = z₂ - z₁
Vector = <x, y, z>
|v| = √(x² + y² + z²)
Component form is one of the most useful ways to write a vector. It shows how far a vector moves along each axis. This makes vector addition, subtraction, graphing, and analysis much easier. Students use component form in algebra, trigonometry, geometry, physics, and engineering.
This component form of a vector calculator turns different inputs into clear vector components. You can start with magnitude and angle. You can also start with two coordinates. If you already know the x and y values, the calculator can analyze that vector too. A 3D option is included for spatial problems.
A vector often appears in polar form first. That means it is given by size and direction. Many math problems become simpler after converting that information into x and y components. The calculator uses cosine and sine for this conversion. It also uses coordinate subtraction when the vector is defined by two points.
Component vectors are used in coordinate geometry, transformation problems, displacement questions, and line analysis. They also help when finding resultant vectors. Once a vector is written as <x, y> or <x, y, z>, it becomes easier to compare directions, find magnitude, and compute unit vectors.
This page does more than give an answer. It shows the formulas and the logical steps. That makes it useful for homework checking and exam revision. Students can see how each part of the vector was formed. Teachers can also use it to prepare quick examples for class discussion.
Manual vector conversion can be slow, especially with decimal angles or negative coordinates. This calculator reduces arithmetic mistakes and saves time. The result section appears above the form for easy review. The export tools also help users save worked examples for later study. For anyone learning vectors, component form is a core skill, and this calculator provides a fast and practical way to master it.
Component form writes a vector by its axis movements. In 2D, that is usually <x, y>. In 3D, it is <x, y, z>. It shows direction and distance along each coordinate axis.
Use x = r cos(θ) and y = r sin(θ). The magnitude is the vector length, and the angle is measured from the positive x-axis unless a problem states otherwise.
Subtract the starting point from the ending point. For 2D, use <x₂ - x₁, y₂ - y₁>. For 3D, also include z₂ - z₁.
atan2(y, x) places the angle in the correct quadrant. That makes it more reliable than basic arctangent when x is negative or when the vector lies on an axis.
Yes. Negative x, y, or z values are valid. They simply indicate direction along the negative side of an axis. The calculator keeps those signs in the final component form.
A unit vector has magnitude 1. It shows direction only. You can find it by dividing each component by the vector magnitude, as long as the vector is not zero.
Use 3D mode when your problem includes x, y, and z coordinates. It is useful for space geometry, physics displacement, and engineering position problems.
Yes. After calculating, use the CSV or PDF buttons. They save the result summary so you can keep a record, share it, or review it later.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.