Three-Dimensional Geometry Questions for NDA

Coordinates and Distance in 3D

Every point in 3D space is written as (x, y, z). The three axes are mutually perpendicular. The distance formula and section formula extend naturally from 2D by just adding the z-component. The midpoint and section formulas work the same way as Class 11 coordinate geometry, just with three components instead of two.

• Distance between P(x1,y1,z1) and Q(x2,y2,z2): sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)
• Midpoint of PQ: ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2)
• Section formula (internal division m:n): ((mx2+nx1)/(m+n), (my2+ny1)/(m+n), (mz2+nz1)/(m+n))
• Origin is (0,0,0); distance from origin to (x,y,z) = sqrt(x^2+y^2+z^2)

Direction Cosines and Direction Ratios

Direction cosines (l, m, n) of a line are the cosines of the angles (alpha, beta, gamma) that the line makes with the positive x, y, z axes respectively. The key identity is l^2 + m^2 + n^2 = 1, always. Direction ratios (a, b, c) are any set of numbers proportional to direction cosines. To convert direction ratios to direction cosines, divide each by sqrt(a^2+b^2+c^2).

• l = cos(alpha), m = cos(beta), n = cos(gamma)
• l^2 + m^2 + n^2 = 1 always (golden identity)
• Direction ratios are proportional to direction cosines, not equal
• DC of x-axis: (1,0,0); y-axis: (0,1,0); z-axis: (0,0,1)
• If line passes through (x1,y1,z1) and (x2,y2,z2), DR = (x2-x1, y2-y1, z2-z1)

Equation of a Line in 3D

A line in 3D is written in two forms: Cartesian and Vector. The standard Cartesian form is the symmetric form, where you equate (x-x1)/a = (y-y1)/b = (z-z1)/c = lambda. Here (x1,y1,z1) is a point on the line and (a,b,c) are direction ratios. Any point on the line can be written as (x1+a*lambda, y1+b*lambda, z1+c*lambda). Vector form is r = a_vec + lambda*b_vec.

• Symmetric (Cartesian) form: (x-x1)/a = (y-y1)/b = (z-z1)/c
• Vector form: r = (x1 i + y1 j + z1 k) + lambda(a i + b j + c k)
• Two-point form: (x-x1)/(x2-x1) = (y-y1)/(y2-y1) = (z-z1)/(z2-z1)
• Angle between two lines: cos(theta) = |l1*l2 + m1*m2 + n1*n2|
• Lines are perpendicular if a1*a2 + b1*b2 + c1*c2 = 0; parallel if a1/a2 = b1/b2 = c1/c2

Equation of a Plane

A plane in 3D is defined by a point and a normal vector. The general equation is ax + by + cz + d = 0, where (a, b, c) is the normal to the plane. Intercept form: x/p + y/q + z/r = 1, where p, q, r are intercepts on x, y, z axes. The plane through three points requires solving a determinant equation. Distance from a point to a plane is one of the most tested formulas.

• General form: ax + by + cz + d = 0; normal vector is (a, b, c)
• Intercept form: x/p + y/q + z/r = 1
• Angle between two planes = angle between their normals
• Planes are parallel if normals are parallel; perpendicular if normals are perpendicular
• Line is parallel to plane if normal of plane is perpendicular to direction of line

Skew Lines and Shortest Distance

Skew lines are lines in 3D that are neither parallel nor intersecting. They exist in different planes. The shortest distance between two skew lines is always along the common perpendicular. For coplanar lines, the shortest distance is zero (they meet or are parallel). NDA sometimes asks you to determine if lines are coplanar by checking the condition using a determinant.

• Skew lines: not parallel and not intersecting; exist in 3D only
• Shortest distance between skew lines formula uses cross product of direction vectors
• If SD = 0, lines are coplanar (either intersecting or parallel)
• Coplanarity condition involves a 3x3 determinant being zero
• For parallel lines, SD = |(point difference) cross (direction)| / |direction|

• Confusing direction cosines with direction ratios — DC must satisfy l^2+m^2+n^2=1; DR are just proportional numbers. Never use raw DR in the angle formula without normalizing.
• Using cos(theta) instead of sin(theta) for angle between a LINE and a PLANE. Remember: line-plane uses sine; line-line or plane-plane uses cosine.
• Forgetting the absolute value in distance from point to plane formula — distance is always positive.
• In the shortest distance formula, mixing up which vector goes in the numerator dot product vs denominator cross product.
• Assuming two lines in 3D that look like they should meet actually do — always verify coplanarity with the determinant condition before concluding lines intersect.

• DC identity: Like a unit vector, DC squares sum to 1. Think of DC as a unit vector dressed in cosine clothes.
• LINE-PLANE angle uses SINE, PLANE-PLANE uses COSINE. Trick: Line is 1D (one letter), Sine also starts differently. Or just remember LP = Sin, PP = Cos.
• For perpendicularity: dot product = 0. For parallelism: ratios equal. DOT for perp, RATIO for parallel.
• Skew lines mnemonic: SKIP — Skew lines are lines that Kross (cross) each other without touching, In 3D, Parallel or intersecting they are Not.

• NDA typically asks 1-2 questions on direction cosines/ratios, 1 on plane equation or distance from point to plane, and 1 on angle between lines or planes. Focus heavily on these.
• Distance of point from plane is a direct plug-and-chug formula — if you know it cold, it is guaranteed marks in under 30 seconds. Memorize it.
• Questions on whether lines are parallel, perpendicular, or skew are common in MCQ format — quickly check dot product and ratio of DRs first.
• Vector form and Cartesian form questions are interchangeable in NDA — practice converting between them so you can pick whichever is faster for the given question.
• Coplanarity and shortest distance between skew lines are slightly harder and appear less frequently — attempt these only after securing easier questions in the section.