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MathidiaMathematics Academy
Further vectors and 3D geometry
Lesson 49 of 75
Challenge

Lines and planes

Objective. Find intersections and classify spatial relationships.

  1. 1. Learn
  2. 2. Worked example
  3. 3. Practice
  4. 4. Feedback
  5. 5. Continue

Learn: the key idea

The relationship between a line and a plane in three dimensions — whether the line is parallel to, lies in, or intersects the plane — is determined by comparing the line's direction vector to the plane's normal vector. If the direction vector is perpendicular to the normal (their dot product is zero), the line is parallel to the plane or lies within it (distinguished by checking if a point on the line satisfies the plane's equation); otherwise the line intersects the plane at exactly one point, found by substituting the line's parametric form into the plane's equation. For example, a line with direction (1,1,0) and a plane with normal (1,−1,2) has dot product 1−1+0=0, so the line is parallel to or lies in the plane.

Worked example

Find the point where the line r = (1,0,2) + t(2,1,−1) meets the plane x + y + z = 10.

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Illustrated card showing 1 … 0
  1. 1Write the line's coordinates: x=1+2t, y=t, z=2−t
  2. 2Substitute into the plane equation: (1+2t)+(t)+(2−t) = 10
  3. 3Simplify: 3+2t = 10, so 2t = 7, t = 7/2
  4. 4Substitute back: x=1+7=8, y=7/2, z=2−7/2=−3/2, giving the point (8, 7/2, −3/2)

Answer: (8, 7/2, −3/2)

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Common mistake. Students find the dot product is zero and immediately conclude the line lies in the plane, without checking that a specific point of the line also satisfies the plane's equation. A zero dot product only shows parallelism, not containment.

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