Vector Direction and Magnitude Tool

Math Linear Algebra • Vectors and Basic Operations

Written by STEM Calculators Team Published February 21, 2026 Updated February 24, 2026

Convert a Cartesian vector to direction–magnitude form: magnitude \(r=\lVert\mathbf{v}\rVert\), angle \(\theta\) in 2D, and spherical angles \((\theta,\phi)\) in 3D. Also compute unit direction vector and direction cosines.

Direction + magnitude visualization

2D: drag to pan • wheel to zoom • double-click to reset. 3D: drag to rotate • Shift+drag to pan • wheel to zoom • double-click to reset. Play animates the arrow growth once and stops.

\(\mathbf{v}\) projection to \(xy\)-plane (3D, dashed) angle guide

Enter a vector and click “Calculate”.

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Theory

Magnitude and direction as a “polar-style” description

Any nonzero vector can be described by a length (its magnitude) and a direction. In Cartesian coordinates we write \(\mathbf{v}=(x,y)\) in 2D or \(\mathbf{v}=(x,y,z)\) in 3D, but many applications prefer a direction–magnitude form because it separates “how big” from “which way.” The magnitude is the Euclidean norm: \[ r=\lVert\mathbf{v}\rVert=\sqrt{x^2+y^2}\quad\text{(2D)},\qquad r=\lVert\mathbf{v}\rVert=\sqrt{x^2+y^2+z^2}\quad\text{(3D)}. \] If \(r=0\) then \(\mathbf{v}=\mathbf{0}\) and the vector has no direction, so angles are undefined.

The unit direction vector is \[ \hat{\mathbf{v}}=\frac{\mathbf{v}}{\lVert\mathbf{v}\rVert}, \] which points in the same direction as \(\mathbf{v}\) but has length 1. Many physics and engineering formulas use \(\hat{\mathbf{v}}\) to represent a direction independent of scale (for example, a force direction or a velocity direction).

2D polar angle

In 2D, we describe direction by an angle \(\theta\) measured from the positive \(x\)-axis. The most reliable way to compute \(\theta\) is the two-argument arctangent: \[ \theta=\operatorname{atan2}(y,x). \] Using \(\operatorname{atan2}\) matters because it automatically chooses the correct quadrant (for example, \(x<0\) places \(\theta\) in the second or third quadrant). A common alternative \(\tan^{-1}(y/x)\) can fail when \(x=0\) and can also lose quadrant information unless additional logic is added.

Once \(\theta\) is known, polar form writes the vector as \[ \mathbf{v}=r\begin{bmatrix}\cos\theta\\ \sin\theta\end{bmatrix}. \] Equivalently, the unit direction vector \(\hat{\mathbf{v}}=(\cos\theta,\sin\theta)\) gives direction cosines in 2D.

3D spherical angles and direction cosines

In 3D, a direction needs two angles. This tool uses the common physics convention:

\(\phi\): azimuth angle in the \(xy\)-plane, measured from \(+x\) toward \(+y\) (\(\phi=\operatorname{atan2}(y,x)\)).
\(\theta\): inclination from the \(+z\)-axis, computed by \(\theta=\cos^{-1}(z/r)\), with \(0^\circ\le\theta\le180^\circ\).

With these definitions, the conversion back to Cartesian coordinates is \[ x=r\sin\theta\cos\phi,\quad y=r\sin\theta\sin\phi,\quad z=r\cos\theta. \] Some textbooks instead use “elevation” from the \(xy\)-plane; that elevation is \(90^\circ-\theta\) in this convention.

The direction cosines in 3D are the cosines of the angles between \(\mathbf{v}\) and the coordinate axes: \[ \cos\alpha=\frac{x}{r},\quad \cos\beta=\frac{y}{r},\quad \cos\gamma=\frac{z}{r}. \] These are exactly the components of the unit vector \(\hat{\mathbf{v}}\). Direction cosines are useful for decomposing a force into axis components, describing orientations in robotics, and interpreting slopes of direction vectors in geometry.

University note: hyperspherical coordinates

In higher dimensions \(\mathbb{R}^n\), a nonzero vector still has a magnitude \(r=\lVert\mathbf{v}\rVert\), but its direction requires multiple angles (a hyperspherical coordinate system). The same theme remains: “radius” plus “angles.” In advanced linear algebra and analysis, you may see these ideas in high-dimensional geometry, optimization, and probability (for example, distributions on the unit sphere).

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