Practice Lines in 3D in Math

Use these practice problems to test your method after reviewing the concept explanation and worked examples.

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Quick Recap

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.

In 2D, a line is defined by a slope and a point (y=mx+by = mx + b). In 3D, slope doesn't work—there's no single number for direction in space. Instead, you specify a starting point and a direction vector (an arrow pointing along the line). The parameter tt acts like a slider: at t=0t = 0 you're at the starting point, and as tt increases or decreases, you slide along the line in the direction of the vector.

Showing a random 20 of 50 problems.

Example 1

hard
Find parametric equations for the line through (1,2,3)(1, 2, 3) perpendicular to both ⟨1,0,0⟩\langle 1, 0, 0 \rangle and ⟨0,1,0⟩\langle 0, 1, 0 \rangle.

Example 2

medium
Determine whether the lines ℓ1:x=1+t,y=2−t,z=3+2t\ell_1: x = 1+t, y = 2-t, z = 3+2t and ℓ2:x=3+2s,y=−s,z=7+4s\ell_2: x = 3+2s, y = -s, z = 7+4s are parallel, intersecting, or skew.

Example 3

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

Example 4

hard
Show that the lines L1:(2+t,1−t,3+t)L_1: (2+t, 1-t, 3+t) and L2:(1+2s,4−2s,2+2s)L_2: (1+2s, 4-2s, 2+2s) are parallel and find a vector connecting them.

Example 5

medium
Do the lines L1:(t,2t,3t)L_1: (t, 2t, 3t) and L2:(1+s,2+s,3+s)L_2: (1+s, 2+s, 3+s) intersect?

Example 6

easy
A line in 3D has direction vector ⟨0,0,1⟩\langle 0, 0, 1 \rangle. The line is parallel to which axis?

Example 7

hard
Find parametric equations of the line through (2,3,5)(2, 3, 5) perpendicular to the plane 2x−y+3z=12x - y + 3z = 1.

Example 8

medium
Find the midpoint of the segment of the line x=t, y=2t, z=tx=t,\ y=2t,\ z=t between t=0t=0 and t=2t=2.

Example 9

challenge
Show that the line L:(1+t,2−t,3+t)L: (1+t, 2-t, 3+t) is parallel to the plane x+2y+z=10x + 2y + z = 10, and find the distance from the line to the plane.

Example 10

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

Example 11

hard
Find the distance between the parallel lines ℓ1:x−12=y1=z+1−1\ell_1: \frac{x-1}{2} = \frac{y}{1} = \frac{z+1}{-1} and ℓ2:x−32=y−11=z−1\ell_2: \frac{x-3}{2} = \frac{y-1}{1} = \frac{z}{-1}.

Example 12

easy
Write the symmetric form of the line through (0,1,2)(0,1,2) with direction ⟨3,4,5⟩\langle 3,4,5 \rangle.

Example 13

easy
Find a point on the line x=2+t, y=−1+2t, z=3tx = 2+t,\ y = -1+2t,\ z = 3t at t=1t = 1.

Example 14

medium
Find the value of tt at which the line x=3t, y=1+t, z=2−tx = 3t,\ y = 1 + t,\ z = 2 - t has z=0z = 0. What point is that?

Example 15

easy
A line has direction ⟨0,1,0⟩\langle 0, 1, 0 \rangle through (2,0,5)(2,0,5). Why can't you write full symmetric form?

Example 16

easy
Write symmetric equations of the line through (2,1,−3)(2, 1, -3) with direction ⟨1,4,2⟩\langle 1, 4, 2 \rangle.

Example 17

challenge
Find the distance from the point (0,0,0)(0,0,0) to the line x=1+t, y=1, z=1x=1+t,\ y=1,\ z=1 (direction ⟨1,0,0⟩\langle1,0,0\rangle).

Example 18

easy
Find the point on the line x=1+2t, y=3−t, z=4tx = 1 + 2t,\ y = 3 - t,\ z = 4t when t=2t = 2.

Example 19

medium
Find where the line x=2+t, y=−1+2t, z=3−tx = 2 + t,\ y = -1 + 2t,\ z = 3 - t crosses the xyxy-plane.

Example 20

medium
Convert symmetric form x−12=y+3−1=z4\frac{x-1}{2} = \frac{y+3}{-1} = \frac{z}{4} to parametric form.
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