Vector Calculator – Complete Guide to Magnitude, Dot Product, Cross Product, Angle and Projection
The Vector Calculator on MyTimeCalculator provides a unified interface for the most common 3D vector operations used in mathematics, physics, engineering, computer graphics and data science. Instead of switching between different tools, you can compute magnitude, unit vectors, dot products, cross products, angles, projections and triple products all in one place.
In many real-world problems, vectors represent quantities that have both magnitude and direction: displacement, velocity, acceleration, force, torque, electric field, magnetic field and more. Understanding how to add vectors, find their direction, resolve one vector along another or compute areas and volumes from them is essential in STEM courses and technical work.
What This Vector Calculator Can Do
This 3D Vector Calculator supports:
- Magnitude and unit vector for a single vector
- Vector addition and subtraction
- Dot product and angle between two vectors
- Cross product and associated area
- Projection of one vector onto another
- Scalar triple product and volume of a parallelepiped
- Vector triple products A × (B × C) and (A × B) × C
All results are computed numerically using standard vector formulas, and formatted for readability with grouped thousands and a configurable number of decimal places.
1. Magnitude and Unit Vector
The magnitude (or length) of a vector A = <Ax, Ay, Az> is given by:
The unit vector in the direction of A is obtained by dividing each component by the magnitude:
Unit vectors are fundamental in physics and engineering: they give direction independently of length, which is useful when specifying directions, axes or orientations in space.
2. Vector Addition and Subtraction
To add or subtract 3D vectors A and B, you work component-wise:
Vector addition and subtraction show up in displacement problems, combining forces, and calculating resultant velocities or accelerations. The calculator returns the resulting vector in bracket form so you can use it directly in your working.
3. Dot Product and Angle Between Vectors
The dot product of two vectors A and B has both geometric and algebraic interpretations:
Here θ is the angle between A and B. Rearranging gives:
The Vector Calculator computes the dot product, the magnitudes of both vectors, and the angle in degrees. This is especially useful when checking if vectors are orthogonal (dot product zero), measuring alignment, or computing work done by a force along a displacement.
4. Cross Product and Area
For 3D vectors, the cross product A × B is defined as:
The cross product is a vector perpendicular to both A and B, with magnitude:
Geometrically, |A × B| is equal to the area of the parallelogram formed by A and B. This makes cross products essential in torque calculations, surface area computations and determining normal vectors for planes and surfaces.
5. Projection of One Vector onto Another
Projecting vector A onto vector B decomposes A into a component parallel to B and a component perpendicular to B. The scalar projection (also called the component of A along B) is:
The vector projection is:
Projection is widely used in resolving forces, shadow lengths, decomposing motion and in numerical methods where you project a vector onto a chosen direction or subspace.
6. Scalar Triple Product and Volume
The scalar triple product of vectors A, B and C is defined as:
Algebraically, it can also be written as the determinant of a 3×3 matrix with rows (or columns) given by the components of A, B and C. Geometrically:
- [A B C] is the signed volume of the parallelepiped formed by A, B and C.
- |[A B C]| is the absolute volume.
- A nonzero value indicates the vectors are not coplanar.
The calculator returns both the signed scalar triple product and its absolute value, helping you analyze volumes in geometry and physics problems.
7. Vector Triple Products
Vector triple products involve the cross product of a vector with a cross product, such as A × (B × C) or (A × B) × C. These expressions appear in vector identities and advanced mechanics.
The calculator directly computes:
- A × (B × C)
- (A × B) × C
In theory, there are identities that can simplify these expressions, but for numerical work it is often easiest to compute them directly using the component formula for cross products. The calculator shows each resulting vector in bracket form for clarity.
How to Use the Vector Calculator
- Choose the operation tab you need (magnitude, dot product, cross product, projection, etc.).
- Enter the components of the vectors in the x, y and z fields.
- Click the corresponding “Compute” button.
- Read the formatted result in the result cards below the button.
If any input is missing or cannot be interpreted as a number, the calculator shows an alert so you can correct the values before running the computation again.
Typical Applications of 3D Vector Calculations
- Physics and engineering: forces, torques, moments, electric fields, magnetic fields.
- Mechanics: rotational motion, angular momentum, equilibrium and stability.
- Computer graphics: surface normals, lighting, camera direction and 3D transformations.
- Robotics: joint motions, workspace analysis and end-effector orientation.
- Navigation: headings, displacement and orientation in 3D space.
- Mathematics and education: checking homework, visualizing lines and planes, exploring vector identities.
Related Tools from MyTimeCalculator
- Dot Product Calculator
- Cross Product Calculator
- Multivariable Calculus Calculator
- Z-Score Calculator
Vector Calculator FAQs
Frequently Asked Questions
Find quick answers 3D vector operations, including magnitude, dot and cross product, angle and projections.
A 3D vector is an ordered triple of numbers that represents magnitude and direction in space. It is usually written as <x, y, z> and can represent displacement, velocity, force, acceleration or any other directional quantity. The Vector Calculator treats each input as a 3D vector and applies standard vector formulas to compute the requested operation.
The angle is computed using the dot product formula: cos(θ) = (A · B) / (|A||B|). The calculator first computes the dot product and magnitudes of both vectors, then uses the arccos function to find θ. Results are shown in degrees, which is convenient for most physics and engineering problems.
A zero dot product indicates that the vectors are orthogonal, or perpendicular to each other in 3D space. This is a useful test for checking right angles, independence of directions and verifying that a vector is normal to a surface or plane.
The cross product A × B is a vector perpendicular to the plane containing A and B. Its magnitude equals the area of the parallelogram formed by the two vectors. In physics, the cross product describes torque, angular momentum and the behavior of particles in magnetic fields, among many other applications.
The scalar projection gives the length of the component of A in the direction of B, possibly with a sign indicating direction. The vector projection gives the actual vector along B that represents this component. The calculator shows both: the scalar projection and the full projection vector projB A.
The scalar triple product [A B C] = A · (B × C) represents the signed volume of the parallelepiped formed by the three vectors. The absolute value |[A B C]| is the actual volume, while the sign indicates orientation. A value of zero means the vectors are coplanar and do not form a 3D volume.
Yes. The Vector Calculator is ideal for checking intermediate and final answers in vector-based physics and engineering problems. You should still show algebraic steps when required, but this tool makes it easy to verify calculations involving dot products, cross products, angles, projections and volumes.
Yes. You can enter any real numbers, including negative values and decimals, for each component. The calculator treats them consistently according to the vector formulas, and it formats results with a convenient number of decimal places. Invalid or missing values trigger an alert so you can correct them.
Some operations, such as computing a unit vector, angle between vectors or projection, are not defined when a vector has zero magnitude. In these cases, the calculator detects the zero length and displays a clear message instead of returning misleading or undefined values.
This specific tool is designed for 3D vectors, which covers most physics, engineering and graphics use cases. If you need higher-dimensional operations, you can still apply the same formulas manually or use dedicated tools that operate on n-dimensional vectors.