Line Meets PlaneJEE Main

Substitute λ, read the pierce point — interactive Mathematics simulation for IIT-JEE.

Concept

A line r = a + λd pierces the plane n · r = p where substitution gives one equation in one unknown λ. If n · d = 0 the line runs parallel to the plane (no solution) — or lies inside it entirely (infinitely many, when a also satisfies the plane). The angle between line and plane is measured from the plane, so it uses sin, not cos.

Key formula

\lambda = \frac{p - \vec n \cdot \vec a}{\vec n \cdot \vec d}; \quad \sin\theta = \frac{|\vec n \cdot \vec d|}{|\vec n||\vec d|}

Derivation

Substitute r = a + λd into n·r = p: n·a + λ(n·d) = p → solve for λ (valid when n·d ≠ 0). Feed λ back into the line for the point.

Angle: d makes angle φ with the normal where cos φ = |n·d|/(|n||d|); the line-plane angle is 90° − φ, hence the sine formula.

Scenarios to explore

Line Meets Plane — Substitute λ, find the pierce point — plus the sin θ angle.

Real-world applications

Ray–surface intersection in computer graphics and ray tracing.
Foot of perpendicular & image of a point in a plane (line along the normal).
Flight-path / terrain-clearance calculations.

JEE exam tips

Image of point P in plane: walk along the normal — image = P + 2·(perp distance)·(∓n̂).
Coplanarity of two lines: [d₁ d₂ (a₂ − a₁)] = 0 — the box-product test.
Distance from parallel line to plane = |n·a − p|/|n| (any point of the line works).

Common mistakes

Using cos instead of sin for the line–plane angle.
Declaring 'no intersection' when n·d = 0 without checking whether the whole line lies in the plane.
Sign slips moving p across the equation.

Exam traps to avoid

λ has units of the direction vector — scaling d rescales λ but NOT the pierce point.
A line 'parallel to the plane' still needs n·a ≠ p; otherwise it's IN the plane.