Line Through a Point (3D) Calculator

This Line Through a Point with a Direction Vector Calculator helps you find the equation of a line in three-dimensional space from a given point and direction vector. Use the point as the starting position and the direction vector to describe how the line moves in the x-, y-, and z-directions. This gives the line in vector form and parametric form. It is a simple way to check answers, understand the method clearly, and practise 3D coordinate geometry step by step.

Step-by-step method

1. Identify the point and direction components.
2. Write the vector form of the line and substitute values.
3. Write the parametric form and substitute values.
4. Write the symmetric form when every direction component is nonzero.

Formulas:

Example 1: (1,2,3),(4,5,6)

Step 1 - Identify the components.

In this problem: Read the point and direction vector components.

Step 2A - Write the vector form.

In this problem: Use r = r₀ + tv.

Step 2B - Substitute values.

In this problem: Replace r₀ with the point and v with the direction vector.

Step 3A - Write the parametric form.

In this problem: Use x = x₀ + at, y = y₀ + bt, z = z₀ + ct.

Step 3B - Substitute values.

In this problem: Substitute x₀, y₀, z₀, a, b, and c into the parametric equations.

Step 3C - Write the symmetric form.

In this problem: Since a, b, and c are nonzero, the symmetric form is valid.

Final answer: Vector form: r = (1, 2, 3) + t(4, 5, 6)

Example 2: (0,0,0),(1,0,0)

Step 1 - Identify the components.

In this problem: Read the point and direction vector components.

Step 2A - Write the vector form.

In this problem: Use r = r₀ + tv.

Step 2B - Substitute values.

In this problem: Replace r₀ with the point and v with the direction vector.

Step 3A - Write the parametric form.

In this problem: Use x = x₀ + at, y = y₀ + bt, z = z₀ + ct.

Step 3B - Substitute values.

In this problem: Substitute x₀, y₀, z₀, a, b, and c into the parametric equations.

Final answer: Vector form: r = (0, 0, 0) + t(1, 0, 0)