Three Dimensional Geometry

🎯 Key Points

• Direction cosines satisfy l²+m²+n²=1 always (they're cosines of angles a UNIT vector makes with the axes); direction ratios are just any proportional set, not normalized
• Lines: perpendicular when b₁·b₂=0 (dot product); parallel when b₁×b₂=0 (cross product) - same pattern as the dot/cross distinction in Vector Algebra
• Shortest distance between skew lines = |(c−a)·(b×d)|/|b×d| - this formula only applies to skew (non-intersecting, non-parallel) lines; intersecting lines have distance 0
• Point-to-plane distance |ax₀+by₀+cz₀+d|/√(a²+b²+c²) directly parallels the 2D point-to-line distance formula, just with one extra coordinate

Three Dimensional Geometry

Extends coordinate geometry to 3D space. Essential for understanding physical space, engineering, and computer graphics.

Direction Cosines and Ratios

• Direction cosines l, m, n: cosines of angles with x, y, z axes
• l² + m² + n² = 1 (fundamental identity)
• Direction ratios a, b, c are proportional to l, m, n
• l = a/√(a²+b²+c²) etc.

Line in 3D

• Vector form: r = a + λb (a: fixed point, b: direction vector)
• Cartesian form: (x−x₁)/a = (y−y₁)/b = (z−z₁)/c
• Angle between lines: cos θ = |b₁·b₂| / (|b₁||b₂|)
• Perpendicular: b₁·b₂ = 0  |  Parallel: b₁ × b₂ = 0

Skew Lines

Non-coplanar, non-intersecting, non-parallel lines in 3D.

Shortest distance = |(c − a) · (b × d)| / |b × d|

Plane

• General form: ax + by + cz + d = 0; normal = (a,b,c)
• Intercept form: x/a + y/b + z/c = 1
• Normal form: lx + my + nz = p

Distances and Angles

• Point to plane: |ax₀+by₀+cz₀+d| / √(a²+b²+c²)
• Angle between planes: cos θ = |n₁·n₂| / (|n₁||n₂|)
• Angle between line and plane: sin θ = |b·n| / (|b||n|)
• Coplanarity of two lines: (c−a)·(b×d) = 0

Coordinates and Distance Formula in 3D

A point in space is fixed by an ordered triple (x, y, z) relative to three mutually perpendicular axes. The three coordinate planes (xy, yz, zx) divide space into eight octants.

• Distance between two points P(x₁,y₁,z₁) and Q(x₂,y₂,z₂): PQ = √[(x₂−x₁)² + (y₂−y₁)² + (z₂−z₁)²].
• Distance from the origin: OP = √(x² + y² + z²).
• Example: distance between (1, −2, 3) and (4, 2, 3) = √[9 + 16 + 0] = 5.

Section Formula in 3D

The point R dividing the segment joining P(x₁,y₁,z₁) and Q(x₂,y₂,z₂) in the ratio m : n has coordinates:

• Internal division: R = ((mx₂ + nx₁)/(m+n), (my₂ + ny₁)/(m+n), (mz₂ + nz₁)/(m+n)).
• External division: replace n by −n, giving ((mx₂ − nx₁)/(m−n), …).
• Midpoint (ratio 1:1): ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2).
• Centroid of a triangle with vertices A, B, C: ((x₁+x₂+x₃)/3, (y₁+y₂+y₃)/3, (z₁+z₂+z₃)/3).

Equation of a Plane - Point-Normal and Three-Point Forms

• Point-normal (vector) form: a plane through point A with normal vector n is n·(r − a) = 0. If n = (A, B, C) and the point is (x₁,y₁,z₁), this expands to A(x−x₁) + B(y−y₁) + C(z−z₁) = 0.
• Plane through three points A, B, C: take n = AB × AC as the normal, then apply the point-normal form. Equivalently the coplanarity determinant of (r−A), (B−A), (C−A) equals 0.
• Distance of the plane from the origin for ax+by+cz+d = 0 is |d|/√(a²+b²+c²).

Plane Through the Intersection of Two Planes

If two planes are P₁: a₁x+b₁y+c₁z+d₁ = 0 and P₂: a₂x+b₂y+c₂z+d₂ = 0, then for any real λ the equation

(a₁x+b₁y+c₁z+d₁) + λ(a₂x+b₂y+c₂z+d₂) = 0

represents a plane through their line of intersection. Choosing λ to satisfy one extra condition (passing through a given point, being perpendicular to another plane, etc.) pins down the required plane without first finding the line of intersection explicitly.

🚀 JEE Advanced Edge

Finding the equation of a plane through three points without memorizing a separate formula: Given three points A, B, C, the plane through them can be found by computing two direction vectors AB and AC, taking their cross product to get the normal vector n=AB×AC, then using the point-normal form n·(r−A)=0 - this reduces "plane through 3 points" to vector operations you already know, rather than a determinant formula to memorize separately.

Why the angle between a line and a plane uses sine, not cosine: The angle between two LINES (or two planes, via their normals) is measured between the direction vectors directly, using cosine - but the angle between a line and a plane is measured between the line and the plane's SURFACE, which is the complement of the angle between the line and the plane's normal, converting the cosine formula into a sine formula: sinθ=|b·n|/(|b||n|).

Worked problem: Find the foot of the perpendicular from the point (1,2,3) to the plane x+y+z=6, and use it to find the distance from the point to the plane. Approach: The plane's normal is (1,1,1). Distance = |1+2+3−6|/√(1²+1²+1²) = |0|/√3 = 0. The point (1,2,3) already lies exactly on the plane (1+2+3=6), so the foot of the perpendicular is the point itself and the distance is 0.

Worked Example: Distance from a Point to a Plane

Find the distance from the point (2, 3, −1) to the plane 2x − y + 2z = 4.

Distance formula: d = |ax₁ + by₁ + cz₁ − d| / √(a² + b² + c²) = |2(2) − (3) + 2(−1) − 4| / √(4+1+4) = |4 − 3 − 2 − 4| / 3 = |−5| / 3 = 5/3 units. The formula always gives a non-negative result - include the absolute value.

Worked Example: Equation of a Line Through Two Points

Find the vector equation of the line passing through A(1, 2, 3) and B(4, 6, 5), and determine a point on it at parameter t = 2.

Direction vector: b = B − A = 3î + 4ĵ + 2k̂. Vector equation: r = (î + 2ĵ + 3k̂) + t(3î + 4ĵ + 2k̂). At t = 2: r = (1+6)î + (2+8)ĵ + (3+4)k̂ = (7, 10, 7). In Cartesian form: (x−1)/3 = (y−2)/4 = (z−3)/2.

Direction Cosines and Direction Ratios of a Line

The direction of a line in space is described by the angles it makes with the coordinate axes.

• Direction cosines l, m, n are the cosines of the angles the line makes with the x, y and z axes respectively.
• They always satisfy the identity l-squared + m-squared + n-squared = 1.
• Direction ratios a, b, c are any numbers proportional to the direction cosines; a line has infinitely many sets of direction ratios but essentially unique direction cosines (up to sign).
• Given direction ratios a, b, c, the direction cosines are a, b, c each divided by the square root of (a-squared + b-squared + c-squared).

Equation of a Line in Space

A line is fixed by a point on it and a direction; this gives both a vector and a cartesian form.

• Vector form (through a point with direction b): r = a + lambda times b, where a is the position vector of a known point and lambda is a scalar parameter.
• Cartesian form: (x - x₁)/a = (y - y₁)/b = (z - z₁)/c, where (x₁, y₁, z₁) lies on the line and a, b, c are direction ratios.
• Line through two points A and B has direction vector b - a, giving r = a + lambda times (b - a).
• If a direction ratio is 0, the corresponding coordinate is constant, so that fraction is replaced by an equation such as x = x₁.

Angle Between Two Lines

The angle between two lines depends only on their directions.

• If the lines have direction vectors b₁ and b₂, then cos theta = |b₁.b₂| divided by (|b₁| times |b₂|).
• In terms of direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂), the same formula uses a₁ times a₂ + b₁ times b₂ + c₁ times c₂ in the numerator.
• The lines are perpendicular when a₁ times a₂ + b₁ times b₂ + c₁ times c₂ = 0.
• The lines are parallel when a₁/a₂ = b₁/b₂ = c₁/c₂, i.e. their direction ratios are proportional.

Angle Between Two Planes and Between a Line and a Plane

Angles involving planes are measured through their normal vectors.

• The angle between two planes equals the angle between their normals: cos theta = |n₁.n₂| divided by (|n₁| times |n₂|).
• Two planes are perpendicular when n₁.n₂ = 0 and parallel when their normals are proportional.
• For a line with direction b and a plane with normal n, the angle phi between the line and the plane satisfies sin phi = |b.n| divided by (|b| times |n|).
• A line is parallel to a plane when b.n = 0, and perpendicular to the plane when b is parallel to n.

Shortest Distance Between Two Lines

Two lines in space may not meet even though they are not parallel; such lines are called skew lines.

• For skew lines r = a₁ + lambda b₁ and r = a₂ + mu b₂, the shortest distance equals |(b₁ cross b₂).(a₂ - a₁)| divided by |b₁ cross b₂|.
• If the shortest distance is 0, the lines intersect (they are coplanar and meet at a point).
• For two parallel lines, the shortest distance is |b cross (a₂ - a₁)| divided by |b|, using the common direction b.
• The shortest distance is always measured along the common perpendicular to both lines.