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Worked solution

Show a parallel line misses a plane

Show that the line

r=(1,1,0)+λ(2,1,4),λ∈R,\mathbf r=(1,1,0)+\lambda(2,1,4),\quad\lambda\in\mathbb R,

does not intersect the plane r⋅(1,2,−1)=4\mathbf r\cdot(1,2,-1)=4.

CJC · Lecture · Example 32

Answer

The line is parallel to and disjoint from the plane.

Compare your reasoning

The line direction is m=(2,1,4)\mathbf m=(2,1,4) and the normal is n=(1,2,−1)\mathbf n=(1,2,-1):

m⋅n=2+2−4=0.\mathbf m\cdot\mathbf n=2+2-4=0.

This shows that the direction is parallel to the plane. Test the line’s base point a=(1,1,0)\mathbf a=(1,1,0):

a⋅n=1+2=3≠4.\mathbf a\cdot\mathbf n=1+2=3\ne4.

Thus the base point is not on the plane. More generally, every line point satisfies (a+λm)⋅n=3+λ(0)=3(\mathbf a+\lambda\mathbf m)\cdot\mathbf n=3+\lambda(0)=3, so no real parameter can satisfy the required value 4. Both the direction and point tests are needed; a zero direction–normal product alone also occurs for a line contained in a plane.

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