Parametric Vector Form Calculator

How the Parametric Vector Form Calculator Works

The Parametric Vector Form Calculator finds the equation of a line in 3D space using two points. This is useful in vector geometry, engineering, physics, computer graphics, robotics, and analytical geometry.

If two points are given:

then the direction vector is:

Parametric Vector Form Formula

The vector form of a line is:

Since v = P₂ − P₁, this becomes:

Scalar Parametric Equations

The same line can be written as three scalar equations:

where v = (a, b, c) is the direction vector.

How to Use the Calculator

1. Enter the coordinates of point P₁.
2. Enter the coordinates of point P₂.
3. Choose the t-range for the 3D plot.
4. Click Calculate Parametric Equation.
5. Review the direction vector, vector form, scalar equations, and plot.

Example Calculation

Suppose:

P₁ = (1, 2, 3)
P₂ = (4, 5, 6)

First calculate the direction vector:

v = P₂ − P₁
v = (4 − 1, 5 − 2, 6 − 3)
v = (3, 3, 3)

The parametric vector form is:

The scalar parametric equations are:

x = 1 + 3t
y = 2 + 3t
z = 3 + 3t

Symmetric Form of a 3D Line

When none of the direction vector components are zero, the line can also be written in symmetric form:

Applications

• Engineering: modeling straight paths, beams, and geometry.
• Physics: representing motion along a straight line.
• Computer graphics: rays, camera paths, and 3D geometry.
• Robotics: path planning and coordinate transformations.
• Mathematics: analytical geometry and vector equations.

Frequently Asked Questions

What is parametric vector form?

Parametric vector form describes every point on a line using a point, a direction vector, and a parameter.

How do I find the direction vector?

Subtract P₁ from P₂: v = P₂ − P₁.

Can the two points be the same?

No. If both points are identical, they do not define a unique line.

What does t mean?

The parameter t controls position along the line. Different t-values produce different points on the same line.

Can this calculator plot the line?

Yes. The calculator uses Plotly to display a 3D line plot over the selected t-range.