Cartesian Vector Calculator

Example Data Table

Formula Used

• Vector form: A = (Ax, Ay, Az), B = (Bx, By, Bz)
• Addition: A + B = (Ax + Bx, Ay + By, Az + Bz)
• Subtraction: A - B = (Ax - Bx, Ay - By, Az - Bz)
• Magnitude: |A| = √(Ax² + Ay² + Az²)
• Unit vector: Â = A / |A|
• 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|²)B
• Scalar triple product: A · (B × C)

How To Use This Calculator

1. Enter x, y, and z components for Vector A.
2. Enter x, y, and z components for Vector B.
3. Enter Vector C when you need the scalar triple product.
4. Enter a scalar value for multiplying vectors.
5. Press the calculate button.
6. Review the result table shown above the form.
7. Use the CSV or PDF button to save your results.

Understanding Cartesian Vectors

A Cartesian vector describes a quantity with components along coordinate axes. In three dimensions, those components are usually x, y, and z. This calculator helps you work with those components without losing the meaning of each step. You can compare two vectors, scale a vector, measure length, find direction, and study products used in geometry, physics, graphics, and engineering problems.

Why Components Matter

Components make vector work organized. Instead of drawing every arrow, you can handle each direction separately. Addition joins matching components. Subtraction compares matching components. Magnitude measures the actual arrow length. A unit vector keeps the direction but changes the length to one. Direction cosines show how much the vector points along each axis.

Products And Geometry

The dot product measures alignment. A positive dot product means the vectors point in a similar general direction. A negative value means they point more opposite. A zero value means perpendicular directions, assuming neither vector has zero length. The cross product creates a new vector perpendicular to both input vectors. Its magnitude equals the area of the parallelogram made by the two vectors. Half of that value gives the triangle area.

Projection And Distance

Projection is useful when one vector must be resolved along another direction. The scalar component tells how far vector A extends along vector B. The projection vector gives the actual component in B direction. The rejection vector shows what remains after that projected part is removed. Distance between endpoints treats the vectors as position points and measures the straight gap between them.

Advanced Uses

The scalar triple product uses three vectors. It measures signed volume for the parallelepiped built from A, B, and C. The sign depends on orientation. The absolute value gives usable volume. These calculations appear in mechanics, 3D modeling, robotics, and coordinate geometry.

Best Practice

Enter values with consistent units. Do not mix meters with centimeters unless you convert first. Review zero vector warnings before using angles or unit vectors. A zero vector has no defined direction. Use the example table to test the form. Then replace sample values with your own components. Export the results when you need records for assignments, reports, or repeated checking across many class scenarios today.

FAQs