What is the difference between a Vector and a Scalar?

A scalar has only magnitude (e.g., mass, temperature). A vector has both magnitude and direction (e.g., velocity, force).

How do you calculate Vector Magnitude?

For a 3D vector (vx, vy, vz), magnitude is calculated using the 3D Pythagorean formula: sqrt(vx^2 + vy^2 + vz^2).

What is a Unit Vector?

A unit vector is a vector normalized to a magnitude of 1, indicating pure direction, calculated as vector divided by its magnitude.

What is the Dot Product of two vectors?

The dot product is a scalar equal to ax*bx + ay*by + az*bz = ||a||*||b||*cos(theta), used to find enclosed angles and work.

What does a Dot Product of zero mean?

If the dot product of two non-zero vectors is zero, the vectors are Orthogonal (perpendicular, 90 degree angle).

What is the Cross Product of two vectors?

The cross product of two 3D vectors yields a new 3D vector perpendicular to both, with magnitude equal to ||a||*||b||*sin(theta).

What is the Right-Hand Rule in cross products?

Curl your right-hand fingers from the first vector toward the second vector; your extended thumb points in the direction of the cross product vector.

What is Vector Projection?

Vector projection calculates the vector component of vector a that acts parallel to reference vector b.

What is the Scalar Triple Product?

The scalar triple product a . (b x c) calculates the volume of the 3D parallelepiped formed by vectors a, b, and c.

How is the Cross Product used in physics?

Cross products calculate rotational physical quantities including torque (r x F), angular momentum (r x p), and magnetic force (q(v x B)).

math calculator

Vector Math & 3D Dot/Cross Product Calculator

Perform 2D/3D vector calculations including magnitude, angle, dot product, and cross product.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Dimensions

Operation

Vector A

Vector B

Results

Result Vector (Rx)

Result Vector (Ry)6

Magnitude |R|7.2111

|A| Magnitude5.0000

|B| Magnitude2.2361

Vector Diagram

A: magnitude 5.000 · 53.1° | B: magnitude 2.236 · 63.4°

Calculation Steps

Add corresponding components of vectors \(\mathbf{A}\) and \(\mathbf{B}\):

\[\mathbf{A}+\mathbf{B} = (3+1,\;4+2) = (4,\;6)\]

Formula

Vector Operation Formulas

\mathbf{A}\cdot\mathbf{B}=\sum A_i B_i, \quad \mathbf{A}\times\mathbf{B}=\det\begin{pmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\A_x&A_y&A_z\\B_x&B_y&B_z\end{pmatrix}

Vectors support addition, subtraction, dot product (scalar result), and cross product (vector result). Use the dot product to find the angle between vectors.

Step by step

How the calculation works

1Enter components for Vector A and Vector B. Select 2D or 3D mode and the operation.
2The diagram shows vectors plotted from the origin with result vector displayed for add/subtract.
3Step-by-step component-wise derivations are shown with LaTeX notation.

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Compute trigonometry, inverse functions, and exponents.

math calculator

Vector Math & 3D Dot/Cross Product Calculator

Perform 2D/3D vector calculations including magnitude, angle, dot product, and cross product.

Formula Verified by Dr. Elena Rostova, PhD

Inputs

Dimensions

Operation

Vector A

Vector B

Results

Result Vector (Rx)

Result Vector (Ry)6

Magnitude |R|7.2111

|A| Magnitude5.0000

|B| Magnitude2.2361

Vector Diagram

A: magnitude 5.000 · 53.1° | B: magnitude 2.236 · 63.4°

Calculation Steps

Add corresponding components of vectors \(\mathbf{A}\) and \(\mathbf{B}\):

\[\mathbf{A}+\mathbf{B} = (3+1,\;4+2) = (4,\;6)\]

Formula

Vector Operation Formulas

\mathbf{A}\cdot\mathbf{B}=\sum A_i B_i, \quad \mathbf{A}\times\mathbf{B}=\det\begin{pmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}\\A_x&A_y&A_z\\B_x&B_y&B_z\end{pmatrix}

Vectors support addition, subtraction, dot product (scalar result), and cross product (vector result). Use the dot product to find the angle between vectors.

Step by step

How the calculation works

1Enter components for Vector A and Vector B. Select 2D or 3D mode and the operation.
2The diagram shows vectors plotted from the origin with result vector displayed for add/subtract.
3Step-by-step component-wise derivations are shown with LaTeX notation.

Related guides

Learn more about this calculation

Guide

Scientific Notation Guide: Powers of Ten Made Practical

Learn how to convert, compare, and calculate with powers of ten using worked examples and the live DTC scientific calculator.

Site Guide

How to Use Do The Calculation Calculators: A Practical Step-by-Step Guide

Learn the fastest reliable workflow for using Do The Calculation calculators, reading results, checking formulas, and using save, print, share, and export actions correctly.

Matrix Solver

Compute determinants, inverses, and solve linear systems.

Scientific

Scientific Calculator

Compute trigonometry, inverse functions, and exponents.