So far you have lived in a flat world of x and y. Three-dimensional geometry adds a third axis, the z-axis, so every point now needs three numbers (x, y, z). With just a handful of formulas — distance, section, direction cosines, lines and planes — you can answer almost any NDA question in this chapter. It is short, formula-driven and a reliable scoring zone.
Stepping Into the Third Dimension
In two dimensions a point is fixed by two numbers, but the real world has depth too. To describe a corner of a room, the tip of a flagpole or the path of an aircraft, we need a third coordinate. So we add a z-axis perpendicular to both the x-axis and y-axis, all meeting at the origin O.
Every point in space is now written as an ordered triple (x, y, z). The three axes split space into eight octants, just as the two axes split a plane into four quadrants.
The three coordinate planes are the XY-plane (z = 0), the YZ-plane (x = 0) and the ZX-plane (y = 0). On the x-axis a point looks like (x, 0, 0); on the y-axis (0, y, 0); on the z-axis (0, 0, z).
For the NDA exam, the most useful habit is to picture which coordinate is zero. If a point lies on a plane, exactly one coordinate is zero; if it lies on an axis, two coordinates are zero. Recognising this instantly removes half the confusion in objective questions.
Distance Between Two Points
The distance formula simply extends the familiar 2D version by adding the squared difference of the z-coordinates.
Distance between P(x1, y1, z1) and Q(x2, y2, z2) is
PQ = √[(x2 − x1)2 + (y2 − y1)2 + (z2 − z1)2].
The distance of a point P(x, y, z) from the origin is simply √(x2 + y2 + z2), since the origin has all coordinates zero.
To check whether three points are collinear, find the three pairwise distances. If the longest equals the sum of the other two, the points lie on one straight line. The same distances also let you test whether a triangle is isosceles, equilateral or right-angled.
Section Formula and Midpoint
To find the point that divides a line segment in a given ratio, extend the 2D section formula coordinate-by-coordinate.
The point dividing the join of P(x1, y1, z1) and Q(x2, y2, z2) internally in ratio m : n is
( (mx2 + nx1)/(m + n), (my2 + ny1)/(m + n), (mz2 + nz1)/(m + n) ).
For external division in ratio m : n, simply replace n with −n (the plus signs become minus). The midpoint is the special case m = n = 1, giving the averages of the coordinates.
The centroid of a triangle with vertices A, B, C is the average of all three: G = ( (x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3 ). NDA tests this directly.
Direction Cosines and Direction Ratios
A line in space points in some direction. The direction cosines capture that direction using the angles the line makes with the three axes.
If a line makes angles α, β, γ with the positive x, y and z axes, then its direction cosines are l = cosα, m = cosβ, n = cosγ.
Direction cosines always satisfy l2 + m2 + n2 = 1. This single identity is one of the most-tested facts in the chapter.
Because a line has two opposite directions, its direction cosines come in two sign-sets: (l, m, n) and (−l, −m, −n). Both describe the same line.
Direction ratios
Direction cosines are tidy but often awkward to compute directly, so we usually work with direction ratios — any set of numbers proportional to the direction cosines.
If a, b, c are direction ratios of a line, the direction cosines are
l = a/√(a2+b2+c2), m = b/√(a2+b2+c2), n = c/√(a2+b2+c2).
For the line joining two points P and Q, a ready set of direction ratios is simply the differences of coordinates: (x2 − x1, y2 − y1, z2 − z1).
Do not confuse direction ratios with coordinates of a point. Direction ratios describe an orientation, while coordinates simply locate a point and obey no normalisation rule.
Two lines with direction ratios (a1, b1, c1) and (a2, b2, c2) are parallel when a1/a2 = b1/b2 = c1/c2, and perpendicular when a1a2 + b1b2 + c1c2 = 0.
Equation of a Straight Line
A straight line in space is fixed by a point on it and a direction. Two standard forms appear in NDA papers.
Line through a point with given direction ratios
If the line passes through (x1, y1, z1) with direction ratios a, b, c, its symmetric (Cartesian) form is
(x − x1)/a = (y − y1)/b = (z − z1)/c.
Line through two points
Through (x1, y1, z1) and (x2, y2, z2):
(x − x1)/(x2 − x1) = (y − y1)/(y2 − y1) = (z − z1)/(z2 − z1).
The common value of the three equal ratios is the parameter. Setting each ratio equal to t gives the handy parametric point (x1 + at, y1 + bt, z1 + ct), useful for finding where a line meets a plane.
Angle Between Two Lines
The angle between two lines depends only on their directions, not on where they are positioned in space.
If the lines have direction ratios (a1, b1, c1) and (a2, b2, c2), the angle θ between them satisfies
cosθ = |a1a2 + b1b2 + c1c2| ÷ [√(a12+b12+c12) · √(a22+b22+c22)].
If you already have direction cosines, the formula is even simpler: cosθ = |l1l2 + m1m2 + n1n2|.
From this single formula two quick tests fall out, and both are NDA favourites: the lines are perpendicular when the numerator (the dot product) is zero, and parallel when their direction ratios are proportional. You almost never need the full angle; usually you just check these two conditions.
Equation of a Plane
A plane is a flat surface. The most useful description uses a vector that sticks straight out of it, called the normal.
The general equation of a plane is ax + by + cz + d = 0, where (a, b, c) are the direction ratios of the normal to the plane.
This is why the coefficients of x, y, z are so important — they instantly tell you the direction the plane faces. Special planes follow at once: the XY-plane is z = 0, the YZ-plane is x = 0, and the ZX-plane is y = 0.
Intercept form
If a plane cuts the axes at distances a, b, c, its equation is x/a + y/b + z/c = 1.
Two planes are parallel if their normals are proportional (same a : b : c) and perpendicular if a1a2 + b1b2 + c1c2 = 0. The same dot-product idea works everywhere in this chapter.
Distance and Angles with a Plane
Two more formulas finish the toolkit and appear nearly every year.
The perpendicular distance of a point P(x1, y1, z1) from the plane ax + by + cz + d = 0 is
D = |ax1 + by1 + cz1 + d| ÷ √(a2 + b2 + c2).
The angle between two planes equals the angle between their normals, so you reuse the line-angle formula with the normal direction ratios.
For the angle between a line and a plane, use sine, not cosine: sinθ = |al + bm + cn| ÷ [√(a2+b2+c2)]. Here (a, b, c) is the plane's normal and (l, m, n) the line's direction. A line is parallel to a plane when this dot product is zero.
Worked Example
Let us combine direction ratios, the angle formula and a perpendicularity check in one typical problem.
Find the distance of the point P(2, 3, −1) from the plane 2x − 3y + 6z + 11 = 0, and state whether the line with direction ratios (3, 2, 0) is parallel to this plane.
So P(2, 3, −1) lies exactly on the plane, giving distance 0, and because the normal is perpendicular to the line's direction, the line is parallel to the plane. This shows how the same dot product answers angle, perpendicularity and parallelism questions all at once.
Mistakes That Cost Marks
3D geometry is forgiving once the formulas are memorised, but small slips drain marks fast. Watch for these.
Using cosine for the angle between a line and a plane. That case needs sine, because the angle is measured from the plane, not from its normal.
- Forgetting the modulus (absolute value) in the distance and angle formulas — distance and the acute angle are never negative.
- Mixing up direction ratios with coordinates of a point.
- Treating the plane's coefficients (a, b, c) as a point instead of the normal direction.
- Dropping a sign when computing differences x2 − x1 for direction ratios.
Previous-Year Style Question
Here is a question in the exact style NDA uses for this chapter.
Q. A line makes equal angles with all three coordinate axes. What is the value of each of its direction cosines?
(a) 1/3 (b) 1/√3 (c) √3 (d) 3
Answer: (b) 1/√3. If the line makes equal angles with each axis, then l = m = n. Using l2 + m2 + n2 = 1, we get 3l2 = 1, so l2 = 1/3 and l = 1/√3. Each direction cosine therefore equals 1/√3 (taking the positive direction).
Quick Revision
Run through this list the night before the exam.
- Distance: PQ = √[(Δx)2 + (Δy)2 + (Δz)2].
- Section (internal, m:n): weighted averages of coordinates.
- Direction cosines: l2 + m2 + n2 = 1.
- Line: (x−x1)/a = (y−y1)/b = (z−z1)/c.
- Plane: ax + by + cz + d = 0; (a, b, c) is the normal.
- Distance of point from plane: |ax1+by1+cz1+d| ÷ √(a2+b2+c2).
- Perpendicular ↔ dot product = 0; parallel ↔ ratios proportional.
Master these seven lines and the entire 3D-geometry section of your NDA paper becomes routine scoring. At The Cavalier, our defence aspirants treat this chapter as a guaranteed-marks zone, because the formulas are few, the dot-product idea repeats everywhere, and a little daily practice locks them into memory. Spend ten focused minutes a day for a week and you will rarely lose a single mark here again.
Frequently asked questions
What are direction cosines of a line?
Direction cosines are the cosines of the angles a line makes with the positive x, y and z axes, written l, m, n. They always satisfy l² + m² + n² = 1, which is the chapter's most tested identity.
What is the difference between direction cosines and direction ratios?
Direction ratios are any numbers proportional to the direction cosines, so they are easier to find from coordinate differences. Dividing each ratio by the square root of the sum of their squares gives the direction cosines.
How do I find the distance of a point from a plane?
For the plane ax + by + cz + d = 0 and point (x₁, y₁, z₁), the distance is |ax₁ + by₁ + cz₁ + d| divided by √(a² + b² + c²). Always keep the modulus so the distance stays non-negative.
Why is sine used for the angle between a line and a plane?
The standard formula gives the angle between the line and the plane's normal using cosine. Since the angle with the plane itself is the complement of that, it comes out as sine of the dot-product expression.
How many marks does 3D Geometry carry in NDA Maths?
Typically 3 to 4 questions appear from this chapter in each NDA Maths paper. Because the topic is short and formula-driven, it offers an excellent marks-per-effort ratio for serious aspirants.
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