3D Geometry
Line and Plane
Grade 12
Question:
<p>If direction ratios of a line are \(l, m, n\) and the direction ratios of normal to a plane are \(a, b, c\), then the condition for the line to be parallel to the plane is:</p>
<p>\(al + bm + cn = 1\)</p>
<p>\(al + bm + cn = 0\)</p>
<p>\(al + bm + cn = -1\)</p>
<p>\(\frac{l}{a} = \frac{m}{b} = \frac{n}{c}\)</p>
Step-by-Step Solution
Key Concept: A line is parallel to a plane if and only if the line is perpendicular to the normal of the plane, which means their direction vectors must be orthogonal (dot product = 0).
Step 1: Recall that a line with direction ratios (l, m, n) is parallel to a plane with normal direction (a, b, c) when the line's direction vector is perpendicular to the plane's normal vector. Step 2: Two vectors are perpendicular if and only if their dot product equals zero. Therefore, the direction vector of the line (l, m, n) and the normal vector (a, b, c) must satisfy: al + bm + cn = 0 Step 3: This condition ensures the line is either parallel to the plane or lies in the plane (both are considered parallel in 3D geometry terminology). ∴ Answer: B (Condition is al + bm + cn = 0 )
Correct Answer: B