The distance between the planes 2x + y − z = 3 and 4x + 2y − 2z = 5 requires you first to:

Divide the second equation by 2 so the coefficients match

Compute the cross product of the normals

The shortest distance between two intersecting lines is:

The distance between their direction vectors

Three-Dimensional Geometry

Three-dimensional geometry is the most systematic topic in the paper, because every object is described by one of two things: a direction for a line and a normal for a plane. Once you know which of those a question gives you, the formula follows without choice. The one genuinely new idea compared with two dimensions is that two lines in space need not meet and need not be parallel — they can be skew, and a large share of the questions exist to test whether you checked for that possibility instead of assuming an intersection.

Lines and Planes

Standard forms in three dimensions
Object	Form	What the numbers mean
Line through a point	(x − x₁)/a = (y − y₁)/b = (z − z₁)/c	⟨a, b, c⟩ are the direction ratios
Line in vector form	r⃗ = a⃗ + λb⃗	a⃗ a point on it, b⃗ its direction
Plane	ax + by + cz + d = 0	⟨a, b, c⟩ is the normal to the plane
Plane in vector form	r⃗·n⃗ = d	n⃗ the normal, d/\|n⃗\| the distance from the origin
Distance from a point to a plane	\|ax₁ + by₁ + cz₁ + d\|/√(a² + b² + c²)	The 2-D formula with one extra term
Direction cosines	l, m, n with l² + m² + n² = 1	Direction ratios divided by their magnitude

🧠 Every angle in 3-D reduces to a dot product between a direction and a normal

There are only three angle questions in this chapter and one idea behind all of them. The angle between two lines is the angle between their direction vectors, so cos θ = |b₁·b₂|/(|b₁||b₂|). The angle between two planes is the angle between their normals, computed the same way. And the angle between a line and a plane is the complement of the angle between the direction and the normal, so sin θ = |b·n|/(|b||n|) — note the sine, not the cosine, because the line makes its angle with the plane rather than with the normal. That single swap of sine for cosine is the most frequently missed detail in the whole topic.

Two consequences follow directly. A line is parallel to a plane when its direction is perpendicular to the normal, that is when b·n = 0, and it lies in the plane if in addition any one of its points satisfies the plane equation. A line is perpendicular to a plane when its direction is parallel to the normal. Note how the conditions invert relative to intuition: parallel to the plane means perpendicular to the normal, which is the source of most sign confusion here.

Skew Lines and Shortest Distances

🎯 Why two lines in space usually miss each other

In a plane two non-parallel lines must meet, because there is nowhere for them to go. In space they can pass over one another entirely — think of two roads at different heights that never cross — and such lines are called skew. So in three dimensions there are three possibilities rather than two: parallel, intersecting, or skew. The test is the scalar triple product [a₂ − a₁, b₁, b₂]: if it is zero the lines are coplanar and therefore either parallel or intersecting, and if it is non-zero they are skew and no point of intersection exists. Solving the equations simultaneously and getting an inconsistency is the same discovery arrived at more slowly, which is why the triple product test is worth doing first.

Shortest-distance formulae
Between	Shortest distance
Two skew lines	\|(a₂ − a₁)·(b₁ × b₂)\| / \|b₁ × b₂\|
Two parallel lines	\|(a₂ − a₁) × b\| / \|b\|
A point and a plane	\|ax₁ + by₁ + cz₁ + d\| / √(a² + b² + c²)
Two parallel planes	\|d₁ − d₂\| / √(a² + b² + c²), after matching coefficients
Intersecting lines	Zero — they meet

⚠️ Match the coefficients before subtracting the constants of two planes

The distance between parallel planes is |d₁ − d₂|/√(a² + b² + c²), but the formula assumes both planes are written with identical coefficients of x, y and z. For 2x + y − z = 3 and 4x + 2y − 2z = 5 you must first divide the second by 2 to get 2x + y − z = 2.5, and only then is the difference of constants meaningful — subtracting 3 and 5 directly gives an answer roughly twice too large. The same normalisation issue appeared with the integrating factor in differential equations, and the lesson generalises: a formula that compares two equations needs them written in the same scale first.

Two constructions complete the chapter. The plane through the intersection of two planes P₁ = 0 and P₂ = 0 is the family P₁ + λP₂ = 0, and imposing one further condition — passing through a given point, or being perpendicular to another plane — fixes λ. And the image of a point in a plane is found by moving from the point along the normal by twice the perpendicular distance, which is also how the foot of the perpendicular is located; the foot is the midpoint of the point and its image, a relation worth using as a check.

Key takeaways

A line is described by a direction and a plane by a normal — identify which the question gives you first.
Direction cosines satisfy l² + m² + n² = 1; direction ratios are any proportional triple.
Angle between lines or between planes uses cosine; angle between a line and a plane uses sine.
A line is parallel to a plane when b·n = 0, and perpendicular when b is parallel to n.
In space two lines may be parallel, intersecting or skew — test with the scalar triple product.
Shortest distance between skew lines is |(a₂ − a₁)·(b₁ × b₂)|/|b₁ × b₂|.
Normalise the coefficients before subtracting constants for the distance between parallel planes.
P₁ + λP₂ = 0 is the family of planes through the intersection of two given planes.

Practice questions (8)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

The angle θ between a line with direction b and a plane with normal n satisfies:
1. cos θ = |b·n|/(|b||n|)
2. sin θ = |b·n|/(|b||n|)
3. tan θ = |b·n|/(|b||n|)
4. θ = 90° always
Show answer
Answer: B — sin θ = |b·n|/(|b||n|)
The dot product gives the angle between the line and the normal, and the angle with the plane itself is the complement of that, so the cosine becomes a sine. Using cosine instead is the single most frequent error in this chapter, and it produces an answer that is wrong by exactly 90° minus the correct value — worth checking whenever a 3-D angle looks implausibly large or small.
Two lines in space that are neither parallel nor intersecting are called:
1. Coplanar
2. Skew
3. Collinear
4. Impossible in three dimensions
Show answer
Answer: B — Skew
Skew lines pass over one another without meeting and without being parallel, a possibility that does not exist in a plane. They are by definition non-coplanar, which is exactly what the scalar triple product test detects: a non-zero value for [a₂ − a₁, b₁, b₂] confirms the lines are skew and no intersection point can be found.
In ax + by + cz + d = 0, the triple ⟨a, b, c⟩ represents:
1. A point on the plane
2. The normal to the plane
3. A direction lying in the plane
4. The distance from the origin
Show answer
Answer: B — The normal to the plane
The coefficients form a vector perpendicular to the plane, which is why the distance formula divides by √(a² + b² + c²) — that is the normal magnitude. Recognising the coefficients as a normal is what makes every angle and parallelism question mechanical: a line parallel to the plane has its direction perpendicular to this triple, and a line perpendicular to the plane has its direction parallel to it.
A line with direction b lies parallel to a plane with normal n when:
1. b is parallel to n
2. b·n = 0
3. b × n = 0
4. |b| = |n|
Show answer
Answer: B — b·n = 0
Being parallel to the plane means the line does not climb out of it, so its direction must be perpendicular to the normal, which is exactly b·n = 0. The conditions invert relative to intuition: b parallel to n, equivalently b × n = 0, makes the line perpendicular to the plane instead. Keeping that inversion in mind removes most of the sign confusion in this topic.
The distance between the planes 2x + y − z = 3 and 4x + 2y − 2z = 5 requires you first to:
1. Subtract 3 from 5 directly
2. Divide the second equation by 2 so the coefficients match
3. Find a point on each plane
4. Compute the cross product of the normals
Show answer
Answer: B — Divide the second equation by 2 so the coefficients match
The formula |d₁ − d₂|/√(a² + b² + c²) is valid only when both planes carry identical coefficients, so the second equation must be scaled to 2x + y − z = 2.5 before the constants are compared. Subtracting 3 from 5 without rescaling gives an answer about twice too large. The cross product of the normals would be zero here, which merely confirms the planes are parallel.
Direction cosines l, m, n of a line always satisfy:
1. l + m + n = 1
2. l² + m² + n² = 1
3. lmn = 1
4. l = m = n
Show answer
Answer: B — l² + m² + n² = 1
Direction cosines are the components of a unit vector along the line, so the sum of their squares is 1. Direction ratios need satisfy no such condition, since any proportional triple describes the same direction — dividing a set of ratios by √(a² + b² + c²) is exactly how the cosines are obtained, and that normalisation is what forces the identity.
The family of planes through the intersection of P₁ = 0 and P₂ = 0 is:
1. P₁ · P₂ = 0
2. P₁ + λP₂ = 0
3. P₁ − P₂ = 0 only
4. P₁ × P₂ = 0
Show answer
Answer: B — P₁ + λP₂ = 0
Any point on the common line satisfies both equations and therefore satisfies P₁ + λP₂ = 0 for every λ, so that expression describes the whole family of planes containing the line. One further condition — passing through a named point, or being perpendicular to a third plane — pins down λ. The special case λ = −1 gives one member of the family, not the family itself.
The shortest distance between two intersecting lines is:
1. Given by the skew-line formula
2. Zero
3. The distance between their direction vectors
4. Undefined
Show answer
Answer: B — Zero
Intersecting lines share a point, so the least distance between them is zero. Applying the skew formula to them would give zero anyway, because the scalar triple product in the numerator vanishes for coplanar lines — a useful consistency check that also shows why a non-zero result from that formula proves the lines are skew.