Lines in 3D Math Example 3

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

Example 3

medium
Determine whether the lines β„“1:x=1+t,y=2βˆ’t,z=3+2t\ell_1: x = 1+t, y = 2-t, z = 3+2t and β„“2:x=3+2s,y=βˆ’s,z=7+4s\ell_2: x = 3+2s, y = -s, z = 7+4s are parallel, intersecting, or skew.

Solution

  1. 1
    Direction vectors: d1βƒ—=⟨1,βˆ’1,2⟩\vec{d_1} = \langle 1, -1, 2 \rangle and d2βƒ—=⟨2,βˆ’1,4⟩\vec{d_2} = \langle 2, -1, 4 \rangle. Check if parallel: 21=2\frac{2}{1} = 2, βˆ’1βˆ’1=1\frac{-1}{-1} = 1. Since 2β‰ 12 \neq 1, they are NOT parallel.
  2. 2
    Check intersection: set 1+t=3+2s1+t = 3+2s, 2βˆ’t=βˆ’s2-t = -s, 3+2t=7+4s3+2t = 7+4s. From equations 1 and 2: t+2s=2t + 2s = 2 and tβˆ’s=2t - s = 2, giving 3s=03s = 0, s=0s = 0, t=2t = 2. Check in equation 3: 3+4=7+03+4 = 7+0, i.e., 7=77 = 7 βœ“. They intersect at (3,0,7)(3, 0, 7).

Answer

TheΒ linesΒ intersectΒ atΒ (3,0,7)\text{The lines intersect at } (3, 0, 7)
In 3D, two lines can be parallel, intersecting, or skew (non-parallel and non-intersecting). To determine which, first check direction vectors for parallelism, then solve the system of parametric equations. If the system has a consistent solution, the lines intersect; otherwise they are skew.

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 xβˆ’x0a=yβˆ’y0b=zβˆ’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 β†’

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 2 medium

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

Example 4 hard

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

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