Lines in 3D Math Example 2

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

Example 2

medium
Find parametric equations for the line through (1,0,−2)(1, 0, -2) and (3,4,1)(3, 4, 1).

Solution

  1. 1
    Find the direction vector: ⟨3−1,4−0,1−(−2)⟩=⟨2,4,3⟩\langle 3-1, 4-0, 1-(-2) \rangle = \langle 2, 4, 3 \rangle.
  2. 2
    Using the point (1,0,−2)(1, 0, -2): x=1+2tx = 1 + 2t, y=4ty = 4t, z=−2+3tz = -2 + 3t.
  3. 3
    The symmetric equations are x−12=y4=z+23\frac{x-1}{2} = \frac{y}{4} = \frac{z+2}{3}.

Answer

x=1+2t,y=4t,z=−2+3tx = 1 + 2t, \quad y = 4t, \quad z = -2 + 3t
When given two points, the direction vector is found by subtracting coordinates. Symmetric equations are an alternative form obtained by solving each parametric equation for tt and setting them equal. Either point can be used as the base point.

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 1 easy

Write parametric equations for the line through the point [formula] in the direction of the vector [

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].

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