Three Dimensional Geometry
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Direct answer
A line needs a point and a direction: (x - x1)/a = (y - y1)/b = (z - z1)/c; a plane needs a point and a normal: a(x - x1) + b(y - y1) + c(z - z1) = 0 with normal vector (a, b, c). Every angle and distance in the chapter reduces to dot and cross products of directions and normals, with direction cosines satisfying l^2 + m^2 + n^2 = 1.
What you must remember
- Direction cosines: for direction ratios (a, b, c), the cosines are a, b, c each divided by sqrt(a^2 + b^2 + c^2), and l^2 + m^2 + n^2 = 1; direction ratios are any proportional triple.
- Line equations: point-direction form as above; through two points, (x - x1)/(x2 - x1) = (y - y1)/(y2 - y1) = (z - z1)/(z2 - z1).
- Plane equations: general ax + by + cz + d = 0 with normal (a, b, c); intercept form x/a + y/b + z/c = 1; through three points by a determinant; the plane through two intersecting lines has normal b1 × b2.
- Angle between two lines: cos theta = |a1a2 + b1b2 + c1c2|/(product of the direction magnitudes); between a line and a plane: sin theta = |direction · normal|/(product of magnitudes); between two planes: the angle between normals.
- Distance from a point to a plane: |ax1 + by1 + cz1 + d|/sqrt(a^2 + b^2 + c^2); between parallel planes |d1 - d2|/sqrt(a^2 + b^2 + c^2) after matching normals.
- Skew lines: the shortest distance is |(a2 - a1)·(b1 × b2)|/|b1 × b2|, where a1, a2 are points and b1, b2 the directions; coplanar exactly when this numerator vanishes.
- Plane family: the plane through the line of intersection of two planes is P1 + lambda P2 = 0, chosen by an extra point or condition.
Common confusion
The angle between a line and a plane uses sine, not cosine — because the angle is measured with the plane's surface, which is the complement of the angle between the line and the normal. Forgetting the absolute value is the second standard loss, since JEE asks for the acute angle. And direction ratios are not direction cosines: (1, 2, 2) are ratios whose cosines are (1/3, 2/3, 2/3).
Exam-focused takeaway
JEE Main asks line-plane angles, point-plane distances, foot of perpendiculars and planes through given points — formula-direct, generous marks if the normal-versus-direction distinction is sharp. JEE Advanced prefers the image of a point in a plane, shortest distance between skew lines with the points of closest approach recovered, coplanarity with parameters, and the family P1 + lambda P2. Write the direction and the normal as vectors first; the rest is dot and cross.
Frequently asked questions
What is the shortest distance between two skew lines?
It is |(a2 - a1)·(b1 × b2)|/|b1 × b2| for points a1, a2 and directions b1, b2; zero means the lines intersect or are parallel.
Why does the line-plane angle use sine?
The angle with the plane complements the angle with its normal, so the dot product with the normal yields sin theta.
How do I test whether two lines are coplanar?
Check that the scalar triple product of (a2 - a1), b1 and b2 is zero; if so, solve for their common point.
What is the plane through the intersection of two planes?
P1 + lambda P2 = 0 — a one-parameter family; fix lambda from an extra point or condition.
What are direction cosines?
The cosines of the angles a line makes with the axes; they satisfy l^2 + m^2 + n^2 = 1, unlike direction ratios.