Formula Used
Magnitude: |A| = √(Ax² + Ay² + Az²).
Unit vector: A / |A|, when |A| is not zero.
Addition: A + B = <Ax + Bx, Ay + By, Az + Bz>.
Subtraction: A - B = <Ax - Bx, Ay - By, Az - Bz>.
Dot product: A · B = AxBx + AyBy + AzBz.
Cross product: A × B = <AyBz - AzBy, AzBx - AxBz, AxBy - AyBx>.
Angle: θ = cos⁻¹[(A · B) / (|A||B|)].
Projection: projB A = [(A · B) / (B · B)]B.
Scalar triple product: A · (B × C).
Vector triple product: A × (B × C).
How to Use This Calculator
Enter the x, y, and z components for vector A. Enter vector B when the operation needs two vectors. Enter vector C for triple product or linear combination work. Add scalar values when needed. Choose the operation. Press submit. The result appears below the header and above the form. Review the formula steps. Use CSV or PDF export for saving the answer.
Example Data Table
| Example |
Vector A |
Vector B |
Operation |
Expected Result |
| 1 |
<3, 4, 0> |
<0, 0, 0> |
Magnitude of A |
5 |
| 2 |
<1, 2, 3> |
<4, 5, 6> |
Dot product |
32 |
| 3 |
<1, 0, 0> |
<0, 1, 0> |
Cross product |
<0, 0, 1> |
| 4 |
<2, 3, 4> |
<1, 1, 1> |
A - B |
<1, 2, 3> |
Why Vector Solving Matters
Vectors describe quantities with size and direction. They appear in geometry, physics, navigation, graphics, robotics, and engineering. A vector can show force, velocity, displacement, acceleration, or any directed change. Because direction matters, normal arithmetic is not enough. Components make vector work clear. Each component tells how much of the vector points along an axis.
What This Calculator Handles
This calculator supports common three dimensional vector tasks. You can find magnitude, unit direction, addition, subtraction, dot product, cross product, angle, projection, distance, scalar multiplication, scalar triple product, and vector triple product. These options cover many classroom and practical problems. The same inputs can be reused for several operations. That makes checking work faster and less error prone.
Reading Vector Results
A magnitude is the length of a vector. A unit vector keeps direction but changes length to one. Addition combines movements or forces. Subtraction compares one vector with another. A dot product measures alignment. A positive dot product shows similar direction. A negative value shows opposite direction. A value near zero shows near perpendicular direction.
Cross Products and Angles
A cross product creates a vector perpendicular to two input vectors. Its magnitude equals the area of the parallelogram formed by them. This is useful for torque, normals, and area calculations. The angle tool uses the dot product and vector lengths. It reports degrees, so the answer is easy to read.
Projection and Advanced Uses
Projection shows how much of one vector lies along another vector. It is helpful when splitting force or motion into useful parts. The scalar triple product gives a signed volume. The vector triple product appears in mechanics and vector identities. With clear steps, you can trace each result and compare it with manual work. Export tools help save answers for reports, assignments, or records. Use consistent units for every component, because the calculator does not convert units automatically.
Good Input Habits
Enter zeros for unused axes when working in two dimensions. Check signs carefully. A small sign change can reverse direction. Keep decimals consistent. Round only after the final answer when precision matters. Review the shown formulas before exporting. They help explain the result to teachers, clients, or teammates in a clear way.
FAQs
What is a vector?
A vector is a quantity with magnitude and direction. It is often written by components, such as <x, y, z>. Common examples include force, velocity, displacement, and acceleration.
Can I use this for two dimensional vectors?
Yes. Enter zero in every z component. The calculator will still process the vectors as three dimensional values, but the unused axis will not affect the result.
What does the dot product show?
The dot product shows alignment between two vectors. Positive values show similar direction. Negative values show opposite direction. A zero value usually means the vectors are perpendicular.
What does the cross product return?
The cross product returns a vector that is perpendicular to both input vectors. Its length equals the parallelogram area formed by those vectors.
Why is my angle not defined?
An angle needs two nonzero vectors. If either vector has zero magnitude, its direction is not defined. The calculator shows a notice in that case.
What is vector projection?
Vector projection shows the part of one vector that lies along another vector. It is useful for splitting motion, force, or displacement into directional components.
What is a scalar triple product?
The scalar triple product is A · (B × C). Its absolute value gives the volume of the parallelepiped formed by the three vectors.
Can I download the results?
Yes. After submitting the form, use the CSV button for spreadsheet records. Use the PDF button for a printable report with the result and steps.