Lines in 3D Math Example 1

Follow the full solution, then compare it with the other examples linked below.

Example 1

easy
Write parametric equations for the line through the point (2,−1,3)(2, -1, 3) in the direction of the vector ⟨4,1,−2⟩\langle 4, 1, -2 \rangle.

Solution

  1. 1
    The parametric equations for a line through (x0,y0,z0)(x_0, y_0, z_0) with direction vector ⟨a,b,c⟩\langle a, b, c \rangle are: x=x0+atx = x_0 + at, y=y0+bty = y_0 + bt, z=z0+ctz = z_0 + ct.
  2. 2
    Substitute: x=2+4tx = 2 + 4t, y=−1+ty = -1 + t, z=3−2tz = 3 - 2t.
  3. 3
    Verify: at t=0t = 0, the point is (2,−1,3)(2, -1, 3) ✓, and the direction is ⟨4,1,−2⟩\langle 4, 1, -2 \rangle ✓.

Answer

x=2+4t,y=−1+t,z=3−2tx = 2 + 4t, \quad y = -1 + t, \quad z = 3 - 2t
A line in 3D is determined by a point and a direction vector. Parametric form represents each coordinate as a linear function of the parameter tt. As tt varies over all real numbers, the equations trace the entire line.

About Lines in 3D

Lines in three-dimensional space described using parametric equations x=x0+atx = x_0 + at, y=y0+bty = y_0 + bt, z=z0+ctz = z_0 + ct, or symmetric form x−x0a=y−y0b=z−z0c\frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c}, where (x0,y0,z0)(x_0, y_0, z_0) is a point on the line and ⟨a,b,c⟩\langle a, b, c \rangle is the direction vector.

Learn more about Lines in 3D →

More Lines in 3D Examples

Example 2 medium

Find parametric equations for the line through [formula] and [formula].

Example 3 medium

Determine whether the lines [formula] and [formula] are parallel, intersecting, or skew.

Example 4 hard

Find the distance between the parallel lines [formula] and [formula].

← All Lines in 3D Examples Lines in 3D Concept