Overview
This chapter introduces three-dimensional (3D) coordinate geometry by extending the familiar two-dimensional Cartesian system to space. Students learn to represent points by ordered triples (x, y, z), compute distances and midpoints in space, and use section (division) formulas. The chapter develops the concepts of direction ratios and direction cosines of a line, and shows how to find angles between lines using scalar (dot) product ideas. Importance: it provides the foundational language and tools for spatial reasoning used later in vectors, 3D calculus, mechanics and geometry problems in higher classes and competitive exams. Key themes include coordinate representation in 3D, metric relations (distance, midpoint), section formula (internal/external division), direction ratios and cosines, and using these to compute angles and lengths. What the student will learn: how to locate and label points in space; apply the distance formula sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2); find midpoints and section points using weighted averages; determine direction ratios (Δx, Δy, Δz) and convert them to direction cosines (l, m, n) with l^2 + m^2 + n^2 = 1; compute the angle between two lines…
Learning Objectives
- Define coordinates of a point in three-dimensional space and represent points by ordered triples (x, y, z).
- State and derive the distance formula between two points in 3‑D and apply it to calculate distances in numerical problems.
- Derive and apply the section formula (internal division) and the midpoint formula for a line segment joining two points.
- Define direction ratios and direction cosines of a line, state the relation between direction cosines, and compute them for given lines.
- Express the vector and Cartesian (symmetric) equations of a line in 3‑D and use them to write the equation of a line through a given point with a specified direction.
- Determine the shortest distance between a point and a line using vector methods and solve related problems.
- Define the equation of a plane in different forms (general, intercept, normal) and derive the connections among these forms.
- Find the equation of a plane passing through three non‑collinear points and the equation of a plane through a point with a given normal vector.
Topics in this chapter
7 topics · tap a topic title to jump straight to it.
Three‑dimensional coordinate system
Definition. A three‑dimensional (3D) rectangular coordinate system is formed by three mutually perpendicular number lines (axes) intersecting at a common origin O. These axes are usually denoted by x, y and z. Every point P in space is represented by an ordered triple P(x, y, z), where x, y and z are the signed perpendicular distances of P from the yz, zx and xy planes respectively.
Axes and planes. The three coordinate planes are: the xy‑plane (z = 0), the yz‑plane (x = 0) and the zx‑plane (y = 0). The intersection of any two coordinate planes is a coordinate axis. The origin O has coordinates (0,0,0).
Octants. Space is divided into eight regions (octants) by the coordinate planes. The first octant consists of points with x > 0, y > 0, z > 0.
Projections. The projection of P(x,y,z) on the xy‑plane is P_xy(x,y,0); similarly on the yz‑plane is (0,y,z) and on zx‑plane is (x,0,z). These projections help visualize 3D positions as corners of a rectangular box (cuboid) with sides |x|, |y|, |z|.
Distance formula (derivation sketch). The distance between P(x1,y1,z1) and Q(x2,y2,z2) is the length of the space diagonal of the cuboid with side lengths |x2-x1|, |y2-y1|, |z2-z1|. By applying the Pythagorean theorem twice: |PQ| = sqrt((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2).
Midpoint and section formulas. The midpoint M of PQ is M( (x1+x2)/2, (y1+y2)/2, (z1+z2)/2 ). For a point dividing PQ internally in the ratio λ:μ (from P to Q), coordinates are ((μ x1 + λ x2)/(λ+μ), (μ y1 + λ y2)/(λ+μ), (μ z1 + λ z2)/(λ+μ)). The external division formula is similar with λ+μ replaced by λ-μ (with sign care).
Direction cosines and ratios. A line through the origin and point P(x,y,z) has direction ratios proportional to (x,y,z). If α, β, γ are angles between the line and the positive x, y, z axes, then direction cosines l = cos α, m = cos β, n = cos γ satisfy l^2 + m^2 + n^2 = 1. For a general line, direction cosines are proportional to its direction ratios.
Common surfaces. Using the distance formula, standard surfaces are described easily: a sphere with center (a,b,c) and radius r is (x-a)^2 + (y-b)^2 + (z-c)^2 = r^2. Coordinate planes are x = constant, y = constant, z = constant (these are vertical or horizontal planes).
Why this matters. The 3D coordinate system provides the algebraic framework to represent points, lines, planes and solids, allowing analytic methods (equations and formulas) to solve geometry, physics and engineering problems.
- Distance between points: Find distance between A(1,2,3) and B(4,6,7). Solution: AB = sqrt((4-1)^2+(6-2)^2+(7-3)^2) = sqrt(3^2+4^2+4^2) = sqrt(9+16+16)=sqrt(41).
- Midpoint: Midpoint of P(2,-1,3) and Q(-4,5,1) is ((2 + (-4))/2, (-1+5)/2, (3+1)/2) = (-1,2,2).
- Internal section: Point R divides A(1,0,0) to B(5,4,2) in ratio 1:2 (A:R:B = 1:2). Coordinates of R = ((2*1 + 1*5)/3, (2*0 + 1*4)/3, (2*0 + 1*2)/3) = (7/3, 4/3, 2/3).
- Direction cosines: For line through origin to P(3, -3, 1) direction cosines are proportional to (3,-3,1). Normalize: l = 3/√(19), m = -3/√(19), n = 1/√(19), and l^2+m^2+n^2 = 1.
- Sphere equation: Sphere with center (1,2,-1) and radius 3 is (x-1)^2 + (y-2)^2 + (z+1)^2 = 9.
- Point: P(x,y,z) represents coordinates along x, y, z axes.
- Distance between two points 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, ratio λ:μ from P to Q): ((μ x1 + λ x2)/(λ+μ), (μ y1 + λ y2)/(λ+μ), (μ z1 + λ z2)/(λ+μ)).
- Direction ratios of line through origin to P(x,y,z): proportional to (x,y,z).
- Direction cosines l,m,n satisfy l = cos α, m = cos β, n = cos γ and l^2 + m^2 + n^2 = 1.
Distance between two points
Definition: In three-dimensional Cartesian coordinates, the distance between two points A(x1, y1, z1) and B(x2, y2, z2) is the length of the straight line segment joining them. It is obtained by applying the Pythagorean theorem to the differences in coordinates.
Derivation (step-by-step):
- Compute the projections onto the coordinate axes: Δx = x2 − x1, Δy = y2 − y1, Δz = z2 − z1.
- Project the segment AB onto the xy-plane to get a horizontal projection of length √(Δx² + Δy²) (Pythagoras in 2D).
- Now consider the right triangle formed by the vertical difference Δz and that horizontal projection. Apply Pythagoras again to get the 3D distance:
- Distance AB = √[(Δx)² + (Δy)² + (Δz)²].
Special cases and interpretations:
- 2D case (z1 = z2): distance reduces to √[(x2 − x1)² + (y2 − y1)²].
- Distance from origin O(0,0,0) to P(x,y,z) is √(x² + y² + z²) — the magnitude (length) of the position vector OP.
- The squared distance, (Δx)² + (Δy)² + (Δz)², is often used to avoid square roots in computations (e.g., comparisons).
How to compute in practice: substitute coordinates into the formula and simplify. Use exact arithmetic when coordinates are integers/rationals; use calculators for decimals.
Note on real-world use: For points on Earth, this straight-line (Euclidean) distance is valid over small scales or when altitude matters. For long distances on Earth's surface, use great-circle (spherical) formulas instead because Earth is curved.
- Numeric example (3D): Find distance between A(1, 2, 3) and B(4, 0, −1). Δx = 3, Δy = −2, Δz = −4. Distance = √(3² + (−2)² + (−4)²) = √(9 + 4 + 16) = √29 ≈ 5.385.
- 2D special case: Distance between P(−1, 3) and Q(2, −1) (take z = 0) = √[(2 − (−1))² + (−1 − 3)²] = √(3² + (−4)²) = √25 = 5.
- Real-life: To find the straight-line distance from a drone at (x1,y1,z1) to a tower top at (x2,y2,z2), compute √[(x2−x1)² + (y2−y1)² + (z2−z1)²], where x,y are horizontal coordinates and z is altitude.
- Distance in 3D: d = √[(x2 − x1)² + (y2 − y1)² + (z2 − z1)²]
- Distance in 2D (z1 = z2): d = √[(x2 − x1)² + (y2 − y1)²]
- Squared distance: d² = (x2 − x1)² + (y2 − y1)² + (z2 − z1)² (useful to avoid square roots)
- Distance from origin: |OP| = √(x² + y² + z²) for P(x,y,z)
- Midpoint (useful related formula): M = ((x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2)
Section formula and midpoint
What is the section formula?
The section formula gives coordinates of a point P that divides the line segment joining two points A(x1, y1, z1) and B(x2, y2, z2) in a given ratio. If P divides AB in the ratio m:n (written AP:PB = m:n), the coordinates of P are obtained by taking the weighted average of coordinates of A and B.
Internal division (AP : PB = m : n)
P = ( (m x2 + n x1)/(m+n),
(m y2 + n y1)/(m+n),
(m z2 + n z1)/(m+n) )
External division (AP : PB = m : n externally)
P = ( (m x2 - n x1)/(m-n),
(m y2 - n y1)/(m-n),
(m z2 - n z1)/(m-n) )
Midpoint
The midpoint M of AB is the special case m = n = 1 (internal), so
M = ( (x1 + x2)/2,
(y1 + y2)/2,
(z1 + z2)/2 )
Vector / parametric viewpoint (useful for proof and understanding)
If A and B are position vectors a and b, points on AB can be written as (1 − t)a + t b. For AP:PB = m:n (internal), choose t = m/(m+n). Substituting gives the section formula above.
How to remember
- For internal division, weights of coordinates are m for the far end (B) and n for the near end (A): (m·B + n·A)/(m+n).
- Midpoint = average of coordinates.
Important remarks
- For internal division, denominator is (m + n); for external, denominator is (m − n) (ensure m ≠ n for external division).
- Works in 2D and 3D; simply ignore z-coordinate in 2D.
- Sign conventions matter: for AP:PB = m:n, the weight m multiplies B's coordinates.
- Example 1 (Internal division): Let A(1, 2, 3) and B(4, 0, -1). Find P that divides AB in the ratio 2:1 (AP:PB = 2:1). Using the formula P = ((2·4 + 1·1)/3, (2·0 + 1·2)/3, (2·(-1) + 1·3)/3) = (9/3, 2/3, 1/3) = (3, 2/3, 1/3).
- Example 2 (Midpoint): Let A(2, -1, 5) and B(-4, 3, 1). Midpoint M = ((2 + (-4))/2, (-1 + 3)/2, (5 + 1)/2) = (-1, 1, 3).
- Example 3 (External division): Let A(1, 0, 0) and B(5, 2, 4). Find point P dividing AB externally in ratio 2:1 (AP:PB = 2:1 external). Use P = ((2·5 - 1·1)/(2 - 1), (2·2 - 1·0)/(2 - 1), (2·4 - 1·0)/(2 - 1)) = (9, 4, 8).
- Internal (AP:PB = m:n): P = ( (m x2 + n x1)/(m + n), (m y2 + n y1)/(m + n), (m z2 + n z1)/(m + n) )
- External (AP:PB = m:n externally): P = ( (m x2 - n x1)/(m - n), (m y2 - n y1)/(m - n), (m z2 - n z1)/(m - n) )
- Midpoint M of A(x1,y1,z1) and B(x2,y2,z2): M = ( (x1 + x2)/2, (y1 + y2)/2, (z1 + z2)/2 )
- Parametric/vector form: Any point on AB can be written as P = (1 - t)A + tB. For AP:PB = m:n (internal), t = m/(m + n).
Direction ratios and direction cosines
Direction ratios (DR)
A line (or a vector) in 3‑D has a direction which can be represented by any triple of numbers (a, b, c) proportional to the components of a direction vector. Such a triple is called a set of direction ratios of the line. If two triples are proportional (one is a nonzero scalar multiple of the other) they represent the same direction.
Direction cosines (DC)
If α, β, γ are the angles made by a line (or vector) with the positive x, y, z axes respectively, then the cosines of these angles l = cosα, m = cosβ, n = cosγ are called the direction cosines. They uniquely determine the direction of a line up to sign, and satisfy
l² + m² + n² = 1.
Relation between direction ratios and direction cosines
If (a, b, c) are direction ratios (i.e. proportional to the components of a direction vector), the corresponding direction cosines are
l = a / √(a² + b² + c²), m = b / √(a² + b² + c²), n = c / √(a² + b² + c²).
Conversely, any set of direction cosines (l, m, n) gives direction ratios proportional to (l, m, n). Note: reversing the direction of the line changes all direction cosines to their negatives.
Angle between two lines
If (l₁, m₁, n₁) and (l₂, m₂, n₂) are direction cosines of two lines, the angle θ between them satisfies
cos θ = l₁l₂ + m₁m₂ + n₁n₂.
Special cases: if cos θ = 0 the lines are perpendicular; if cos θ = ±1 they are parallel (same or opposite direction).
Lines not through origin
Direction ratios/cosines depend only on the direction (vector) of the line. For a line through points P(x₁,y₁,z₁) and Q(x₂,y₂,z₂), direction ratios are (x₂−x₁, y₂−y₁, z₂−z₁).
- Example 1: Given direction ratios (2, -3, 6). Compute direction cosines. Normalize: √(2² + (-3)² + 6²) = √(4+9+36)=√49=7. So l = 2/7, m = -3/7, n = 6/7. Check: l²+m²+n² = (4+9+36)/49 = 1.
- Example 2: Angle between lines with direction ratios (1,2,2) and (2,-1,2). First convert to direction cosines by normalization: for first vector magnitude √(1+4+4)=3 so (1/3,2/3,2/3); for second magnitude √(4+1+4)=3 so (2/3,-1/3,2/3). Then cosθ = (1/3)(2/3)+(2/3)(-1/3)+(2/3)(2/3) = (2-2+4)/9 = 4/9. So θ = cos⁻¹(4/9).
- Example 3: Determine if lines through P(1,0,2)→Q(3,4,5) and R(0,1,1)→S(2,5,4) are parallel. Direction ratios PQ = (2,4,3), RS = (2,4,3). They are proportional (equal) so the lines are parallel (same direction).
- Direction cosines from direction ratios (a,b,c): l = a/√(a²+b²+c²), m = b/√(a²+b²+c²), n = c/√(a²+b²+c²).
- Normalization identity: l² + m² + n² = 1.
- Direction ratios from two points P(x₁,y₁,z₁), Q(x₂,y₂,z₂): (a,b,c) = (x₂−x₁, y₂−y₁, z₂−z₁).
- Angle between two lines with direction cosines (l₁,m₁,n₁) and (l₂,m₂,n₂): cosθ = l₁l₂ + m₁m₂ + n₁n₂.
- Perpendicular condition: l₁l₂ + m₁m₂ + n₁n₂ = 0. Parallel condition: direction ratios are proportional.
Angle between two lines
In three-dimensional geometry the angle between two straight lines is defined as the angle between their direction vectors. If two lines are given by their direction vectors a = (a1, a2, a3) and b = (b1, b2, b3), the angle θ between the lines is the angle between these vectors. The value of θ is taken as the acute angle between them (0 ≤ θ ≤ 90°) unless the oriented angle is required.
Common line representations and how direction is read:
- Vector form: r = r0 + t a — direction vector is a.
- Parametric form: x = x1 + a1 t, y = y1 + a2 t, z = z1 + a3 t — direction ratios are (a1,a2,a3).
- Symmetric form: (x − x1)/l = (y − y1)/m = (z − z1)/n — direction ratios are (l,m,n).
Derivation (brief): The cosine of the angle between two vectors a and b is given by the dot product formula cos θ = (a · b)/(|a| |b|). Therefore the cosine of the angle between the two lines equals the cosine between their direction vectors. If cos θ is negative, the acute angle is 180° − θ; equivalently use |cos θ| to get the acute angle directly.
Special cases: parallel lines have θ = 0° or 180° (direction vectors proportional). Perpendicular lines have θ = 90° (dot product = 0). For skew lines (non-parallel, non-intersecting) the angle is still defined by their direction vectors.
- Example 1 — Symmetric forms: Given L1: (x − 1)/2 = (y + 1)/(−1) = (z − 2)/3 and L2: (x + 2)/1 = (y − 3)/4 = (z + 1)/(−2). Direction ratios: a = (2, −1, 3), b = (1, 4, −2). Compute a·b = 2·1 + (−1)·4 + 3·(−2) = −8. |a| = √(4+1+9)=√14, |b| = √(1+16+4)=√21. cos θ = −8/(√14·√21) = −8/√294 ≈ −0.4664. So θ ≈ arccos(−0.4664) ≈ 117.8°, and the acute angle = 180° − 117.8° ≈ 62.2°.
- Example 2 — Vector form through origin: Let lines have direction vectors a = (1,2,2) and b = (2,−1,1). Then a·b = 1·2 + 2·(−1) + 2·1 = 2. |a| = √(1+4+4)=3, |b| = √(4+1+1)=√6 ≈ 2.449. cos θ = 2/(3·√6) ≈ 0.2722, so θ ≈ arccos(0.2722) ≈ 74.2°.
- Example 3 — Perpendicular test: If direction vectors are a = (1,0,0) and b = (0,1,0), then a·b = 0 so cos θ = 0 and θ = 90°. Thus the lines are perpendicular.
- If direction vectors are a = (a1,a2,a3) and b = (b1,b2,b3): cos θ = (a · b) / (|a| |b|) = (a1 b1 + a2 b2 + a3 b3) / (√(a1^2+a2^2+a3^2) · √(b1^2+b2^2+b3^2)).
- Using direction ratios (l1,m1,n1) and (l2,m2,n2): cos θ = (l1 l2 + m1 m2 + n1 n2) / (√(l1^2+m1^2+n1^2) · √(l2^2+m2^2+n2^2)).
- If lines have slopes m1 and m2 in a plane (2D case): tan θ = |(m2 − m1)/(1 + m1 m2)|. (Use this only for 2D.)
- Parallel condition: direction ratios proportional ⇒ θ = 0° or 180°. Perpendicular condition: a · b = 0 ⇒ θ = 90°.
- To get the acute angle use θ_ac = arccos(|(a·b)| / (|a||b|)) or if cos θ given, take θ_ac = min(θ, 180° − θ).
Equations of a line in three dimensions
Overview
A line in 3D is the set of points that extends in one direction determined by a fixed point and a fixed direction. In three-dimensional Cartesian coordinates a point is (x, y, z). To describe a line we need one point on it and a direction vector.
Vector form
If A(x1, y1, z1) is a point on the line and b = <l, m, n> is a direction vector, the vector equation is:
r = a + λb
where r = <x, y, z> is the position vector of a general point on the line, a = <x1, y1, z1> and λ is a real parameter.
Parametric form
From r = a + λb we get three parametric equations:
x = x1 + λl, y = y1 + λm, z = z1 + λn.
Symmetric (Cartesian) form
Eliminating the parameter λ gives the symmetric form (if l, m, n are nonzero):
(x - x1)/l = (y - y1)/m = (z - z1)/n.
If any of l, m or n is zero, write the corresponding coordinate directly (for example if l = 0 then x = x1 and the other two ratios are equal).
Two-point form
A line through two distinct points A(x1,y1,z1) and B(x2,y2,z2) has direction vector b = <x2-x1, y2-y1, z2-z1> and vector/parametric/symmetric equations obtained by substituting b and a = A.
Direction cosines and ratios
If l, m, n are direction cosines of the line then l^2 + m^2 + n^2 = 1. Direction ratios are any proportional numbers to the direction cosines (i.e., if p:q:r are direction ratios then (p, q, r) is parallel to the direction vector).
Relative positions of two lines
Let lines have direction vectors b1 and b2.
- Parallel: b1 is proportional to b2 (direction ratios proportional).
- Intersecting: there exist parameters λ, μ such that positions are equal (solve parametric equations and get a common point).
- Skew: not parallel and do not intersect (no common solution).
Line as intersection of two planes
The intersection of two non-parallel planes (P1: a1x + b1y + c1z + d1 = 0 and P2: a2x + b2y + c2z + d2 = 0) is a line. Its direction vector is perpendicular to normals of both planes, i.e., b = n1 × n2.
Remarks for visualization
Think of a 3D arrow: its tail at point A and arrowhead showing the direction vector b. Project the line onto the xy-, yz-, and zx-planes to see traces (straight lines) that help sketch it.
- 1) Line through A(1,2,3) and B(4,0,5). Direction vector b = <4-1, 0-2, 5-3> = <3, -2, 2>. Vector form: r = <1,2,3> + λ<3,-2,2>. Parametric: x = 1 + 3λ, y = 2 - 2λ, z = 3 + 2λ. Symmetric: (x-1)/3 = (y-2)/(-2) = (z-3)/2.
- 2) Line through P(2,-1,3) parallel to direction vector <1,2,-1>. Vector form: r = <2,-1,3> + t<1,2,-1>. Parametric: x = 2 + t, y = -1 + 2t, z = 3 - t.
- 3) Determine relation of L1: r = <0,0,0> + s<1,2,3> and L2: r = <1,-1,2> + t<2,4,6>. Since <2,4,6> = 2<1,2,3>, direction vectors are proportional so lines are parallel. Check if they coincide by seeing if a point of L2 lies on L1: solve <1,-1,2> = s0<1,2,3> gives s0=1, 2*1=2 (not -1), so no. Thus L1 and L2 are distinct parallel lines.
- Vector form: r = a + λb, where a = <x1,y1,z1> and b = <l,m,n>
- Parametric form: x = x1 + λl, y = y1 + λm, z = z1 + λn
- Symmetric (Cartesian) form: (x - x1)/l = (y - y1)/m = (z - z1)/n (if l,m,n ≠ 0)
- Two-point direction vector: b = <x2-x1, y2-y1, z2-z1>
- Direction cosines l,m,n satisfy l^2 + m^2 + n^2 = 1 (if normalized)
- Line as intersection of planes P1 and P2: direction b = n1 × n2 (cross product of normals)
Applications and simple results
Overview. In three dimensional coordinate geometry (3D), every point is given by an ordered triple (x, y, z) relative to mutually perpendicular axes OX, OY, OZ. Many basic results—distance, section, midpoint, line equations and direction measures—are extensions of 2D formulas using the 3D Pythagorean theorem. These results are widely used in physics, engineering, graphics and navigation.
1. Distance between two points. For points P(x1, y1, z1) and Q(x2, y2, z2), distance PQ = sqrt((x2−x1)2 + (y2−y1)2 + (z2−z1)2). This follows by applying Pythagoras first in the xy-plane and then with the z-difference.
2. Section (division) and midpoint. A point dividing segment AB internally in ratio m:n has coordinates ((nx1 + mx2)/(m+n), (ny1 + my2)/(m+n), (nz1 + mz2)/(m+n)). For external division replace n by −n. The midpoint is the special case m=n=1 giving ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2).
3. Direction ratios and direction cosines. A line in 3D has direction ratios (a, b, c) if any vector parallel to it is proportional to (a, b, c). If l, m, n are direction cosines (cosines of angles made with OX, OY, OZ) then l2 + m2 + n2 = 1 and they are proportional to the direction ratios. If a vector is v = (a, b, c) and |v| = sqrt(a2+b2+c2), then direction cosines = (a/|v|, b/|v|, c/|v|).
4. Equation of a line. Vector form: r = r0 + t u where r0 is position vector of a point on the line and u = (a, b, c) is the direction vector. Parametric form: x = x0 + at, y = y0 + bt, z = z0 + ct. Symmetric form (when a, b, c ≠ 0): (x − x0)/a = (y − y0)/b = (z − z0)/c.
5. Simple results and tests. - Collinearity of three points A, B, C: vectors AB and AC are proportional (or area of triangle = 0). Equivalently the direction ratios from A→B and A→C are proportional. - Centroid of a triangle with vertices A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3) is ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3). - Shortest distance between two skew (non-parallel, non-intersecting) lines r = a + t u and r = b + s v is |( (b − a) · (u × v) )| / |u × v|. This gives the length of the common perpendicular.
Applications (brief). Calculating straight-line distances in 3D (e.g., between two floors in a building), locating midpoints for construction or design, finding points that divide beams or routes in given ratios, determining orientation of rods/antennas via direction cosines, computing shortest distance between non-intersecting paths (important in robotics and collision avoidance), and plotting lines and planes in 3D modeling and computer graphics.
- Distance: Find distance between P(1, 2, 3) and Q(4, 0, −1). Compute sqrt((4−1)^2 + (0−2)^2 + (−1−3)^2) = sqrt(3^2 + (−2)^2 + (−4)^2) = sqrt(9+4+16) = sqrt(29).
- Midpoint: Midpoint of A(2, −1, 5) and B(8, 3, 1) is ((2+8)/2, (−1+3)/2, (5+1)/2) = (5, 1, 3).
- Section formula: Point dividing AB internally in ratio 2:1 where A(0,0,0), B(6,3,9) is ((1*0 + 2*6)/(2+1), (1*0 + 2*3)/3, (1*0 + 2*9)/3) = (4, 2, 6).
- Direction cosines: For vector v = (2, −3, 6), |v| = sqrt(4+9+36) = sqrt(49) = 7. Direction cosines = (2/7, −3/7, 6/7). Check: (2/7)^2 + (−3/7)^2 + (6/7)^2 = 1.
- Equation of a line: Line through P(1,0,2) and Q(3,4,5): direction vector = Q−P = (2,4,3). Symmetric form: (x−1)/2 = (y−0)/4 = (z−2)/3.
- Distance between P(x1,y1,z1) and Q(x2,y2,z2): sqrt((x2−x1)^2 + (y2−y1)^2 + (z2−z1)^2).
- Internal section (m:n) of A(x1,y1,z1) and B(x2,y2,z2): ((n x1 + m x2)/(m+n), (n y1 + m y2)/(m+n), (n z1 + m z2)/(m+n)).
- Midpoint of A and B: ((x1+x2)/2, (y1+y2)/2, (z1+z2)/2).
- Direction cosines l,m,n of vector (a,b,c): l = a/√(a^2+b^2+c^2), m = b/√(a^2+b^2+c^2), n = c/√(a^2+b^2+c^2) and l^2 + m^2 + n^2 = 1.
- Vector equation of a line: r = r0 + t(a,b,c). Parametric: x = x0 + at, y = y0 + bt, z = z0 + ct. Symmetric (if a,b,c ≠ 0): (x−x0)/a = (y−y0)/b = (z−z0)/c.
- Centroid of triangle with vertices A,B,C: ((x1+x2+x3)/3, (y1+y2+y3)/3, (z1+z2+z3)/3).
Key Concepts
- Three-dimensional coordinate system
- A system with three mutually perpendicular axes (x, y, z) meeting at the origin to locate any point by an ordered triple (x, y, z).
- Origin
- The fixed point where the three axes meet, denoted by O with coordinates (0, 0, 0).
- Coordinate axes
- The three perpendicular lines labeled x-axis, y-axis and z-axis used to measure coordinates in 3D space.
- Point (x, y, z)
- An ordered triple representing the position of a point in 3D measured along x, y and z axes from the origin.
- Distance between two points
- Distance between P(x1,y1,z1) and Q(x2,y2,z2) is sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1)^2).
- Midpoint
- Midpoint of segment joining P(x1,y1,z1) and Q(x2,y2,z2) is ((x1+x2)/2,(y1+y2)/2,(z1+z2)/2).
- Section formula (internal division)
- If a point divides AB with A(x1,y1,z1), B(x2,y2,z2) internally in ratio m:n (AP:PB = m:n), its coordinates are ((mx2+nx1)/(m+n),(my2+ny1)/(m+n),(mz2+nz1)/(m+n)).
- Direction ratios (DRs)
- A triplet of numbers proportional to the components of a direction vector of a line; e.g., (l, m, n) such that any scalar multiple represents the same direction.
- Direction cosines
- Cosines of angles that a line (or vector) makes with the x, y and z axes, usually denoted (l, m, n) satisfying l^2 + m^2 + n^2 = 1.
- Vector
- A quantity with magnitude and direction represented in 3D as v = (vx, vy, vz) or as position difference between two points.
- Magnitude (norm) of a vector
- Length of vector v = (vx,vy,vz) is |v| = sqrt(vx^2 + vy^2 + vz^2).
- Unit vector
- A vector of length 1 in the direction of a given vector v, found by v/|v|.
- Dot product (scalar product)
- For u=(u1,u2,u3) and v=(v1,v2,v3), u·v = u1v1 + u2v2 + u3v3; equals |u||v|cosθ.
- Cross product (vector product)
- For u and v in R^3, u×v is a vector perpendicular to both with magnitude |u||v|sinθ and direction given by right-hand rule.
- Equation of a line (parametric & symmetric forms)
- Parametric: x = x1 + lt, y = y1 + mt, z = z1 + nt where (l,m,n) is direction vector. Symmetric: (x-x1)/l = (y-y1)/m = (z-z1)/n when l,m,n ≠ 0.
- Angle between two lines
- If direction vectors are u and v, the angle θ between lines satisfies cosθ = (u·v)/(|u||v|).
- Skew lines
- Two lines in 3D that are neither parallel nor intersecting (do not lie in the same plane).
- Equation of a plane (general form)
- A plane is given by ax + by + cz + d = 0 where (a,b,c) are constants not all zero.
- Normal vector to a plane
- A vector perpendicular to the plane; for ax + by + cz + d = 0 the normal is (a, b, c).
- Distance of a point from a plane
- Distance from P(x0,y0,z0) to plane ax+by+cz+d=0 is |ax0+by0+cz0+d| / sqrt(a^2+b^2+c^2).
End-of-Chapter Trial Paper & Test Questions
Topic-wise questions to test your understanding of every concept in this chapter.
-
How many octants divide three-dimensional space, and what are the signs of the coordinates in the first octant? / त्रिविमीय आकाश को कितने अष्टांश विभाजित करते हैं, तथा प्रथम अष्टांश में निर्देशांकों के चिह्न क्या होते हैं?
Show answer
The three coordinate planes divide space into eight octants; in the first octant all coordinates are positive (x > 0, y > 0, z > 0). / तीन निर्देशांक तल आकाश को आठ अष्टांशों में विभाजित करते हैं; प्रथम अष्टांश में सभी निर्देशांक धनात्मक होते हैं (x > 0, y > 0, z > 0)।
-
Find the distance between the points A(1, 2, 3) and B(4, 6, 7). / बिंदुओं A(1, 2, 3) और B(4, 6, 7) के बीच की दूरी ज्ञात कीजिए।
Show answer
AB = √((4−1)² + (6−2)² + (7−3)²) = √(9 + 16 + 16) = √41 units. / AB = √((4−1)² + (6−2)² + (7−3)²) = √(9 + 16 + 16) = √41 इकाई।
-
Find the coordinates of the point dividing the segment joining A(1,0,0) and B(5,4,2) internally in the ratio 1:2. / A(1,0,0) और B(5,4,2) को मिलाने वाले रेखाखंड को 1:2 के अनुपात में आंतरिक रूप से विभाजित करने वाले बिंदु के निर्देशांक ज्ञात कीजिए।
Show answer
Using the section formula with m:n = 1:2, R = ((1·5+2·1)/3, (1·4+2·0)/3, (1·2+2·0)/3) = (7/3, 4/3, 2/3). / m:n = 1:2 के साथ विभाजन सूत्र से, R = ((1·5+2·1)/3, (1·4+2·0)/3, (1·2+2·0)/3) = (7/3, 4/3, 2/3)।
-
Find the direction cosines of the line whose direction ratios are (2, −3, 6). / उस रेखा की दिक्-कोज्याएँ ज्ञात कीजिए जिसकी दिक्-अनुपात (2, −3, 6) हैं।
Show answer
The magnitude is √(4+9+36) = √49 = 7, so the direction cosines are l = 2/7, m = −3/7, n = 6/7, satisfying l² + m² + n² = 1. / परिमाण √(4+9+36) = √49 = 7 है, अतः दिक्-कोज्याएँ l = 2/7, m = −3/7, n = 6/7 हैं, जो l² + m² + n² = 1 को संतुष्ट करती हैं।
-
State the relation satisfied by the direction cosines of a line and explain its geometric meaning. / किसी रेखा की दिक्-कोज्याओं द्वारा संतुष्ट संबंध बताइए तथा उसका ज्यामितीय अर्थ समझाइए।
Show answer
The direction cosines satisfy l² + m² + n² = 1, meaning the squares of the cosines of the angles the line makes with the three axes always add to one (the unit direction vector has length 1). / दिक्-कोज्याएँ l² + m² + n² = 1 को संतुष्ट करती हैं, जिसका अर्थ है कि रेखा द्वारा तीनों अक्षों के साथ बनाए कोणों की कोज्याओं के वर्गों का योग सदैव एक होता है (एकक दिशा सदिश की लंबाई 1 है)।
-
Find the angle between the lines with direction ratios (1, 2, 2) and (2, −1, 2). / दिक्-अनुपात (1, 2, 2) तथा (2, −1, 2) वाली रेखाओं के बीच का कोण ज्ञात कीजिए।
Show answer
Both vectors have magnitude 3, so cos θ = (1·2 + 2·(−1) + 2·2)/(3·3) = (2−2+4)/9 = 4/9, giving θ = cos⁻¹(4/9). / दोनों सदिशों का परिमाण 3 है, अतः cos θ = (1·2 + 2·(−1) + 2·2)/(3·3) = (2−2+4)/9 = 4/9, जिससे θ = cos⁻¹(4/9)।
-
Write the symmetric (Cartesian) form of the line through P(1, 0, 2) and Q(3, 4, 5). / बिंदुओं P(1, 0, 2) और Q(3, 4, 5) से गुजरने वाली रेखा का सममित (कार्तीय) रूप लिखिए।
Show answer
The direction vector is Q − P = (2, 4, 3), so the symmetric form is (x−1)/2 = y/4 = (z−2)/3. / दिशा सदिश Q − P = (2, 4, 3) है, अतः सममित रूप (x−1)/2 = y/4 = (z−2)/3 है।
-
How can you check whether two lines in 3D are parallel using their direction ratios? / दिक्-अनुपातों का प्रयोग करके यह कैसे जाँचा जाए कि त्रिविम में दो रेखाएँ समांतर हैं?
Show answer
Two lines are parallel if and only if their direction ratios are proportional (one set is a non-zero scalar multiple of the other); equivalently cos θ = ±1 between their direction vectors. / दो रेखाएँ समांतर होती हैं यदि और केवल यदि उनके दिक्-अनुपात समानुपाती हों (एक समुच्चय दूसरे का अशून्य अदिश गुणज हो); समतुल्य रूप से उनके दिशा सदिशों के बीच cos θ = ±1।
Related Laws & Principles
Explore allFoundational laws & principles behind this chapter. Each one opens a full page — what it says, why it matters, five practice questions and the mistakes to avoid.