Three-dimensional geometry for NDA

Three-dimensional geometry sounds harder than it is. Almost every result is a two-dimensional formula with a z-term added, and most NDA questions test one result at a time: a distance, a set of direction cosines, an angle, a plane's intercepts, a sphere's centre. The syllabus covers points, direction cosines and ratios, lines and planes in various forms, angles between lines and between planes, and the sphere. If you are comfortable with vectors, you already know most of this chapter under a different name.

Typical question types are:

• the distance between two points, or of a point from an axis or plane;
• the ratio in which a coordinate plane divides a segment;
• direction cosines from direction ratios, and the l² + m² + n² = 1 condition;
• the third angle a line makes with the axes, given two;
• the angle between two lines or two planes, and when they are parallel or perpendicular;
• the equation of a plane from its intercepts, and the distance between parallel planes;
• the centre and radius of a sphere.

Points in space

The three coordinate planes divide space into eight octants. The xy-plane is z = 0, the yz-plane is x = 0 and the zx-plane is y = 0.

• Distance between (x₁, y₁, z₁) and (x₂, y₂, z₂): √[(x₂ − x₁)² + (y₂ − y₁)² + (z₂ − z₁)²].
• Distance of (x, y, z) from the x-axis: √(y² + z²). Similarly √(z² + x²) from the y-axis and √(x² + y²) from the z-axis.
• Section formula (internal, m : n): ((mx₂ + nx₁)/(m + n), (my₂ + ny₁)/(m + n), (mz₂ + nz₁)/(m + n)).
• Centroid of a triangle: the average of the three vertices, coordinate by coordinate.

To find the ratio in which a coordinate plane divides a segment, take the ratio as k : 1 and set the relevant coordinate to zero. A negative k means external division.

Direction cosines and direction ratios

If a line makes angles α, β and γ with the positive x-, y- and z-axes, its direction cosines are l = cos α, m = cos β and n = cos γ. They always satisfy l² + m² + n² = 1, and so sin²α + sin²β + sin²γ = 2.

The reason: a unit vector along the line has components (l, m, n), and its length is 1.

Direction ratios a, b, c are any three numbers proportional to l, m, n. To convert, divide by √(a² + b² + c²): l = a/√(a² + b² + c²), and similarly for m and n. The line through (x₁, y₁, z₁) and (x₂, y₂, z₂) has direction ratios (x₂ − x₁, y₂ − y₁, z₂ − z₁).

Angle between two lines

For lines with direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂):

cos θ = |a₁a₂ + b₁b₂ + c₁c₂| ÷ [√(a₁² + b₁² + c₁²) × √(a₂² + b₂² + c₂²)].

With direction cosines the denominator is 1, so cos θ = |l₁l₂ + m₁m₂ + n₁n₂|.

• Perpendicular: a₁a₂ + b₁b₂ + c₁c₂ = 0.
• Parallel: a₁/a₂ = b₁/b₂ = c₁/c₂.

Lines and planes

Line through (x₁, y₁, z₁) with direction ratios a, b, c: (x − x₁)/a = (y − y₁)/b = (z − z₁)/c.

Angle between two planes is the angle between their normals: cos θ = |a₁a₂ + b₁b₂ + c₁c₂| ÷ (product of the normals' lengths). Planes are perpendicular when a₁a₂ + b₁b₂ + c₁c₂ = 0, and parallel when the normals are proportional.

Angle between a line and a plane: if the line has direction ratios (a, b, c) and the plane has normal (A, B, C), then sin φ = |aA + bB + cC| ÷ (product of the lengths). It is sin, not cos, because φ is the complement of the angle with the normal.

Distances:

• point (x₁, y₁, z₁) to plane ax + by + cz + d = 0: |ax₁ + by₁ + cz₁ + d|/√(a² + b² + c²);
• between parallel planes ax + by + cz + d₁ = 0 and ax + by + cz + d₂ = 0: |d₁ − d₂|/√(a² + b² + c²), after making the coefficients identical.

The sphere

x² + y² + z² + 2ux + 2vy + 2wz + d = 0 has centre (−u, −v, −w) and radius √(u² + v² + w² − d). It is the circle's general equation with one more variable.

Worked NDA-style MCQs

Q1. A line makes angles of 45° and 60° with the x- and y-axes. The angle it makes with the z-axis is:
(a) 30° only (b) 60° or 120° (c) 45° (d) 90°

cos²γ = 1 − cos²45° − cos²60° = 1 − 1/2 − 1/4 = 1/4. So cos γ = ±1/2 and γ = 60° or 120°. Answer: (b).

Q2. The direction cosines of the line joining (1, 2, 3) and (3, 5, 9) are:
(a) 2/7, 3/7, 6/7 (b) 2, 3, 6 (c) 1/7, 2/7, 3/7 (d) 2/11, 3/11, 6/11

Direction ratios (2, 3, 6), and √(4 + 9 + 36) = 7. Answer: (a).

Q3. The angle between lines with direction ratios (1, 1, 2) and (√3 − 1, −√3 − 1, 4) is:
(a) 30° (b) 45° (c) 60° (d) 90°

Dot product: (√3 − 1) − (√3 + 1) + 8 = 6. The lengths are √6 and √[(4 − 2√3) + (4 + 2√3) + 16] = √24 = 2√6. cos θ = 6/(√6 × 2√6) = 6/12 = 1/2. Answer: (c).

Q4. The distance between the planes 2x − y + 2z + 3 = 0 and 4x − 2y + 4z + 5 = 0 is:
(a) 1/6 (b) 1/3 (c) 2/3 (d) 8/3

Halve the second: 2x − y + 2z + 5/2 = 0. Distance = |3 − 5/2|/√(4 + 1 + 4) = (1/2)/3 = 1/6. Answer: (a).

Q5. The plane cutting intercepts 2, 3 and 4 on the axes is:
(a) 2x + 3y + 4z = 1 (b) 6x + 4y + 3z = 12 (c) 4x + 3y + 2z = 24 (d) 6x + 4y + 3z = 24

x/2 + y/3 + z/4 = 1; multiply by 12. Answer: (b).

Q6. The centre and radius of the sphere x² + y² + z² − 2x + 4y − 6z − 2 = 0 are:
(a) (1, −2, 3), 4 (b) (−1, 2, −3), 4 (c) (1, −2, 3), 16 (d) (2, −4, 6), 4

u = −1, v = 2 and w = −3, so the centre is (1, −2, 3). Radius √(1 + 4 + 9 + 2) = 4. Answer: (a).

Common mistakes

• Treating direction ratios as direction cosines. Divide by the length first.
• Forgetting the ± in cos γ, which gives two possible angles.
• Using cos for the angle between a line and a plane. It is sin, because the normal is involved.
• Unequal coefficients in the parallel-planes formula. Scale first, exactly as for parallel lines.
• Sign slips in the sphere's centre. The centre is (−u, −v, −w), half the coefficients with the sign flipped.

Practice set

1. The distance of (2, 3, 6) from the origin is: (a) 11 (b) 7 (c) 49 (d) √11
2. The direction cosines for direction ratios 1, 2, 2 are: (a) 1, 2, 2 (b) 1/5, 2/5, 2/5 (c) 1/3, 2/3, 2/3 (d) 1/9, 2/9, 2/9
3. The distance of the origin from x + 2y + 2z = 6 is: (a) 6 (b) 3 (c) 2 (d) 1
4. The xy-plane divides the join of (1, 2, 3) and (4, −1, −6) in the ratio: (a) 1 : 2 internally (b) 2 : 1 internally (c) 1 : 2 externally (d) 3 : 6 externally
5. The distance of (3, 4, 5) from the x-axis is: (a) 3 (b) 5 (c) √41 (d) √34
6. The angle between the planes 2x − y + z = 6 and x + y + 2z = 7 is: (a) π/6 (b) π/4 (c) π/3 (d) π/2
7. Lines with direction ratios (2, k, 3) and (1, −2, 4) are perpendicular when k is: (a) −7 (b) 7 (c) 5 (d) 1
8. The centroid of the triangle with vertices (1, 2, 3), (4, 5, 6) and (−2, −1, 0) is: (a) (1, 2, 3) (b) (3, 6, 9) (c) (2, 2, 3) (d) (1, 3, 2)

Answers:

1. (b). √(4 + 9 + 36) = 7.
2. (c). √(1 + 4 + 4) = 3.
3. (c). 6/√(1 + 4 + 4) = 2.
4. (a). With ratio k : 1, z = (−6k + 3)/(k + 1) = 0 gives k = 1/2.
5. (c). √(4² + 5²).
6. (c). cos θ = |2 − 1 + 2|/(√6 × √6) = 3/6 = 1/2.
7. (b). 2 − 2k + 12 = 0.
8. (a). (3/3, 6/3, 9/3).

What to do next

• Write the plane-forms table and the three angle formulas from memory.
• Solve 25 old NDA questions from this chapter, and for each angle question note whether it needed cos or sin.
• Study vectors alongside this chapter, and compare the formulas with their two-dimensional versions in straight lines.