Free Vector Calculator

Comprehensive Vector Operations

A vector calculator simplifies tasks like vector addition, subtraction, dot product, cross product, projection, normalization, magnitude calculation, and finding the vector between two points. It works for both 2‑dimensional (plane) and 3‑dimensional vectors, accepting input as Cartesian coordinates or (for plane vectors) magnitude and direction.

Representing Vectors in Cartesian Coordinates

In a plane, a vector is described by two numbers: $\mathbf{a} = [a_x, a_y]$, where $a_x$ is the horizontal displacement and $a_y$ the vertical displacement. In space, a third component is added: $\mathbf{a} = [a_x, a_y, a_z]$.

Magnitude and Direction

Instead of components, a vector can be specified by its magnitude $m$ (also called the norm) and its direction angle $\theta$ relative to the positive x‑axis. The conversion between the two representations is given by:

$$a_x = m\cos\theta,\quad a_y = m\sin\theta.$$

The magnitude can be recovered from components using the Pythagorean theorem:

$$m = \sqrt{a_x^2 + a_y^2} \quad \text{or} \quad m = \sqrt{a_x^2 + a_y^2 + a_z^2} \ \text{in 3D}.$$

Vector Addition and Subtraction

Adding vectors is performed component‑wise:

$$\mathbf{a} + \mathbf{b} = [a_x + b_x,\ a_y + b_y,\ a_z + b_z].$$

For example, with $\mathbf{a}=[2,3,4]$ and $\mathbf{b}=[1,-2,3]$:

$$\mathbf{a}+\mathbf{b} = [3,1,7].$$

Subtraction follows the same pattern:

$$\mathbf{a} - \mathbf{b} = [a_x - b_x,\ a_y - b_y,\ a_z - b_z],$$

which gives $[1,5,1]$ for the same vectors. Graphically, addition obeys the parallelogram rule; subtraction is equivalent to adding the opposite of $\mathbf{b}$.

Vector Between Two Points

To obtain the vector that goes from point $P$ to point $Q$, subtract $P$’s coordinates from $Q$’s: $\overrightarrow{PQ} = [Q_x-P_x,\ Q_y-P_y,\ Q_z-P_z]$. This operation is essentially a vector subtraction.

Dot Product (Scalar Product)

The dot product of two vectors returns a scalar defined by:

$$\mathbf{a}\cdot\mathbf{b} = |\mathbf{a}|\,|\mathbf{b}|\,\cos\theta,$$

where $\theta$ is the angle between them. In Cartesian coordinates:

$$\mathbf{a}\cdot\mathbf{b} = a_x b_x + a_y b_y + a_z b_z.$$

Using $\mathbf{a}=[2,3,4]$ and $\mathbf{b}=[1,-2,3]$,

$$\mathbf{a}\cdot\mathbf{b} = 2\cdot1 + 3\cdot(-2) + 4\cdot3 = 8.$$

The dot product is commutative and can be used to find the angle between vectors:

$$\cos\theta = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|}.$$

Also, $\mathbf{a}\cdot\mathbf{a} = |\mathbf{a}|^2$.

Cross Product (Vector Product)

The cross product produces a vector perpendicular to both $\mathbf{a}$ and $\mathbf{b}$. Its magnitude equals the area of the parallelogram they span:

$$|\mathbf{a}\times\mathbf{b}| = |\mathbf{a}|\,|\mathbf{b}|\,\sin\theta.$$

The direction is given by the right‑hand rule. In Cartesian coordinates:

$$\mathbf{a}\times\mathbf{b} = \bigl[ a_y b_z - a_z b_y,\ a_z b_x - a_x b_z,\ a_x b_y - a_y b_x \bigr].$$

For $\mathbf{a}=[2,3,4]$ and $\mathbf{b}=[1,-2,3]$:

$$\mathbf{a}\times\mathbf{b} = [3\cdot3 - 4\cdot(-2),\ 4\cdot1 - 2\cdot3,\ 2\cdot(-2) - 3\cdot1] = [17, -2, -7].$$

The cross product is anti‑commutative: $\mathbf{b}\times\mathbf{a} = -\mathbf{a}\times\mathbf{b}$.

Vector Projection

The projection of $\mathbf{b}$ onto $\mathbf{a}$ is the component of $\mathbf{b}$ that lies along $\mathbf{a}$. It is computed as:

$$\operatorname{proj}_{\mathbf{a}}\mathbf{b} = \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|^{2}}\,\mathbf{a}.$$

With the example vectors:

$$\operatorname{proj}_{\mathbf{a}}\mathbf{b} = \frac{8}{29}\,[2,3,4] = \left[\frac{16}{29},\frac{24}{29},\frac{32}{29}\right].$$

Geometrically, this is the “shadow” of $\mathbf{b}$ on $\mathbf{a}$.

Vector Normalization and Magnitude

Normalizing a vector means scaling it to a length of 1 while preserving its direction. First compute the norm (magnitude):

$$|\mathbf{a}| = \sqrt{a_x^2 + a_y^2 + a_z^2}.$$

Then divide each component by the norm:

$$\mathbf{\hat{u}} = \frac{\mathbf{a}}{|\mathbf{a}|} = \left[\frac{a_x}{|\mathbf{a}|},\frac{a_y}{|\mathbf{a}|},\frac{a_z}{|\mathbf{a}|}\right].$$

For $\mathbf{a}=[2,3,4]$, $|\mathbf{a}|=\sqrt{29}$, so the normalized vector is $\left[\frac{2}{\sqrt{29}},\frac{3}{\sqrt{29}},\frac{4}{\sqrt{29}}\right]$.

Using the Calculator

Start by selecting whether you are working in 2D or 3D. Then choose the desired operation: addition, subtraction, dot product, cross product, projection, normalization, magnitude, or vector between points. Enter your data either as Cartesian components or as magnitude and direction (for 2D). The calculator instantly computes the result and, where applicable, shows the intermediate formulas.

This vector operation calculator combines multiple tools—a vector addition calculator, dot product calculator, cross product calculator, normalization calculator, projection calculator, and magnitude calculator—into one convenient interface, making it a versatile resource for learning and problem solving.