Unit 11
Analytical Geometry
Class 12 Mathematics
Co-ordinates in Space
Class 12 Mathematics – Co-ordinates in Space Notes PDF
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NEB / CDC Focus
This unit develops three-dimensional analytical geometry: distance between points in space, direction cosines and direction ratios of a line, equations of planes in several forms, planes determined by points or by the intersection of two planes, conditions for parallel and perpendicular planes, angle between planes, and distance from a point to a plane.
1. Coordinates in Three-Dimensional Space
In space, a point is located by an ordered triple (x,y,z). The three coordinate axes are mutually perpendicular: the x-axis, y-axis and z-axis.
A point P(x,y,z) has perpendicular coordinates x, y and z measured relative to the three coordinate planes.
| Coordinate plane | Equation |
|---|---|
| xy-plane | z=0 |
| yz-plane | x=0 |
| zx-plane | y=0 |
2. Distance Between Two Points in Space
Let P(x₁,y₁,z₁) and Q(x₂,y₂,z₂). By extending the Pythagorean theorem to three dimensions,
Distance of P(x,y,z) from the origin is
Find the distance between P(1,2,3) and Q(4,6,6).
PQ=√[(4−1)²+(6−2)²+(6−3)²]=√(9+16+9)=√34.
3. Direction Cosines and Direction Ratios of a Line
Suppose a directed line makes angles α,β,γ with the positive x, y and z axes respectively.
The numbers l=cosα, m=cosβ, n=cosγ are called the direction cosines of the line.
Any three numbers a,b,c proportional to the direction cosines are called direction ratios of the line.
If a,b,c are direction ratios, then
For the line through P(x₁,y₁,z₁) and Q(x₂,y₂,z₂), convenient direction ratios are
Find direction cosines corresponding to direction ratios 2,−3,6.
√(2²+(-3)²+6²)=√49=7.
l=2/7, m=−3/7, n=6/7.
4. General Equation of a Plane
The general equation of a plane is Ax+By+Cz+D=0, where A,B,C are not all zero.
The vector (A,B,C) is normal to the plane. Thus A,B,C are direction ratios of the plane’s normal.
Find the plane through (1,2,−1) with normal direction ratios (2,−1,3).
2(x−1)−(y−2)+3(z+1)=0
2x−y+3z+3=0.
5. Intercept Form and Normal Form of a Plane
5.1 Intercept Form
If a plane cuts the x, y and z axes at (a,0,0), (0,b,0), and (0,0,c), then
5.2 Normal Form
If the perpendicular from the origin to the plane has length p and makes angles α,β,γ with the positive axes, then
With direction cosines l,m,n of the normal, this is
6. Equation of a Plane Through Three Given Points
Three non-collinear points determine a unique plane. If the points are (x₁,y₁,z₁), (x₂,y₂,z₂) and (x₃,y₃,z₃), the plane may be written as
x₂−x₁ y₂−y₁ z₂−z₁ ;
x₃−x₁ y₃−y₁ z₃−z₁ | = 0.
Assume Ax+By+Cz+D=0, substitute all three points, and solve the resulting equations for a proportional set of coefficients.
Find the plane through (1,0,0), (0,1,0), (0,0,1).
These are intercepts 1,1,1, so the plane is
x+y+z=1.
7. Plane Through the Intersection of Two Given Planes
Let the two planes be
P₁≡A₁x+B₁y+C₁z+D₁=0
P₂≡A₂x+B₂y+C₂z+D₂=0.
Every plane passing through their line of intersection is represented by
where λ is a real parameter.
A further condition—such as passing through a given point, being parallel to another plane, or being perpendicular to another plane—is then used to determine λ.
Find the plane through the intersection of x+y+z−1=0 and 2x−y+z−3=0 that passes through (1,1,1).
The required family is
(x+y+z−1)+λ(2x−y+z−3)=0.
Substitute (1,1,1):
2+λ(−1)=0 ⇒ λ=2.
Therefore
x+y+z−1+2(2x−y+z−3)=0
5x−y+3z−7=0.
8. Parallel and Perpendicular Planes
Consider
A₁x+B₁y+C₁z+D₁=0,
A₂x+B₂y+C₂z+D₂=0.
8.1 Parallel Planes
Planes are parallel when their normal vectors are parallel:
If the full coefficient ratios including constants are also equal, the two equations represent the same coincident plane.
8.2 Perpendicular Planes
Planes are perpendicular when their normal vectors are perpendicular:
Show that x+2y−z+4=0 and 2x−y+3=0 are perpendicular.
Normals are (1,2,−1) and (2,−1,0).
1(2)+2(−1)+(−1)(0)=0.
Hence the planes are perpendicular.
9. Angle Between Two Planes
The angle between two planes is defined as the acute or specified angle between their normal vectors.
Find the angle between x+y+z=0 and x−y=0.
Normals: (1,1,1) and (1,−1,0).
dot=1−1+0=0.
Therefore θ=90°; the planes are perpendicular.
10. Distance of a Point from a Plane
For the point P(x₁,y₁,z₁) and plane Ax+By+Cz+D=0, the perpendicular distance is
Find the distance of (1,2,3) from 2x−y+2z−5=0.
d=|2(1)−2+2(3)−5|/√(4+1+4) =|1|/3 =1/3.
11. More Worked Examples
For P(1,−1,2) and Q(4,3,−2), direction ratios of PQ are
4−1 : 3−(−1) : −2−2 = 3:4:−4.
Find the plane cutting the axes at 2,3,6.
x/2+y/3+z/6=1.
Multiplying by 6:
3x+2y+z=6.
Find a plane through (2,−1,3) parallel to 2x+3y−z+5=0.
It has the same normal:
2(x−2)+3(y+1)−(z−3)=0.
2x+3y−z+2=0.
Find k if x+2y+kz=0 is perpendicular to 2x−y+2z=0.
1(2)+2(−1)+k(2)=0 ⇒2k=0 ⇒k=0.
Convert 2x−2y+z−9=0 to normal form.
Magnitude of normal coefficients: √(4+4+1)=3.
(2/3)x−(2/3)y+(1/3)z=3.
Thus the perpendicular distance from origin is p=3.
Find the plane through (2,0,0), (0,4,0) and (0,0,8).
x/2+y/4+z/8=1 ⇒4x+2y+z=8.
12. Problem-Solving Strategy
- In 3D distance problems, subtract corresponding coordinates before squaring.
- For direction cosines, normalize direction ratios by √(a²+b²+c²).
- Read (A,B,C) immediately as a normal vector to Ax+By+Cz+D=0.
- For a plane through one point with known normal, use point-normal form directly.
- Use intercept form only when the three axis intercepts are known and non-zero.
- For a plane through three points, use the determinant form or solve the general-plane coefficients.
- For a plane through the intersection of two planes, begin with P₁+λP₂=0.
- Test parallelism using proportional normals; test perpendicularity using zero dot product.
- Use the same normal vectors in the angle formula.
- For point-to-plane distance, substitute the point in the plane expression and divide by the normal magnitude.
13. Common Mistakes and Warnings
In space, do not forget the (z₂−z₁)² term.
Direction ratios must be normalized before being called direction cosines.
For Ax+By+Cz+D=0, the normal is (A,B,C), not including D.
Parallelism depends on A,B,C ratios; D need not have the same ratio unless the planes coincide.
Use the dot product of normals; do not multiply slopes as in elementary 2D geometry.
Point-to-plane distance must be non-negative, so the numerator uses absolute value.
A plane through the intersection of P₁=0 and P₂=0 is P₁+λP₂=0.
Three collinear points do not determine a unique plane.
14. Exam-Important Formula Sheet
15. Important Exam Questions
Short-Answer Questions
- Write the distance formula for two points in space.
- Define direction cosines and direction ratios of a line.
- Prove/state l²+m²+n²=1.
- Write the general equation of a plane and identify its normal.
- State the intercept form and normal form of a plane.
- State the family of planes through the intersection of two given planes.
- State the conditions for two planes to be parallel and perpendicular.
- Write the formula for angle between two planes.
- Write the formula for distance of a point from a plane.
Derivation / Long Questions
- Derive the distance formula between two points in space.
- Derive the relation l²+m²+n²=1 for direction cosines.
- Obtain the intercept form of a plane.
- Obtain the normal form of a plane.
- Find the equation of a plane through three given non-collinear points.
- Derive/use the equation of a plane through the intersection of two planes.
- Derive the angle formula using the normals of two planes.
Numerical Questions
- Find distance between two points in space.
- Find direction cosines from given direction ratios.
- Find a plane through a point with a specified normal.
- Find a plane from its three intercepts.
- Find a plane through three given points.
- Find a plane through the intersection of two planes satisfying an extra condition.
- Find a parameter so that two planes are parallel or perpendicular.
- Find the angle between two planes.
- Find the perpendicular distance of a point from a plane.
Diagram Questions
- Draw the 3D coordinate axes with a point P(x,y,z).
- Illustrate direction angles of a line with the coordinate axes.
- Draw a plane and its normal.
- Draw a plane with x-, y- and z-intercepts.
- Draw two intersecting planes and their common line.
- Illustrate the perpendicular distance of a point from a plane.
16. One-Minute Revision
- A point in space is written as (x,y,z).
- 3D distance uses the squared changes in x, y and z.
- Direction cosines are cosα, cosβ, cosγ.
- They satisfy l²+m²+n²=1.
- Direction ratios are any proportional triple to the direction cosines.
- For Ax+By+Cz+D=0, normal direction ratios are A,B,C.
- Point-normal plane: A(x−x₁)+B(y−y₁)+C(z−z₁)=0.
- Intercept form is x/a+y/b+z/c=1.
- Normal form is lx+my+nz=p.
- Three non-collinear points determine a unique plane.
- Planes through a common intersection line are P₁+λP₂=0.
- Parallel planes have proportional normals.
- Perpendicular planes have normal dot product zero.
- The angle between planes is found from the angle between their normals.
- Point-to-plane distance needs an absolute value in the numerator.
17. Diagram Practice
- Three-dimensional x, y and z coordinate axes.
- 3D displacement components between two points.
- Direction angles and direction cosines of a line.
- General plane with a normal vector.
- Plane in intercept form.
- Family of planes through the intersection of two planes.
- Angle between two planes using normals.
- Perpendicular distance from a point to a plane.
Also Visit
Original Nepal eNotes page: Class 12 Mathematics Co-ordinates in Space Notes.
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