Planes and Lines
Given the plane
Determine if the line given by the parametric equations
Is perpendicular, parallel, or neither to the plane.
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1 solution
The normal vector of the plane is given by ⟨ 1 , 2 , 3 ⟩ and the direction vector of the line is given by ⟨ 1 , 4 , − 3 ⟩
Therefore,
⟨ 1 , 2 , 3 ⟩ ⋅ ⟨ 1 , 4 , − 3 ⟩ = 0
Which implies that the normal vector of the plane is normal to to the direction vector of the line, from which we can deduce that the line is parallel to the plane.
Note :A simple plug in of t = 0 gives us that the line indeed does not lie in the plane.