Unit 4: 3D Geometry - Lines and Spheres

Shortest Distance Between Two Lines

Skew Lines

In 3D geometry, two lines that are neither parallel nor intersecting are called **skew lines**. The shortest distance between them is the length of the line segment that is perpendicular to both lines.

Shortest Distance Formula (Cartesian)

Let the two skew lines L₁ and L₂ be given by:

L₁: (x - x₁)/l₁ = (y - y₁)/m₁ = (z - z₁)/n₁
L₂: (x - x₂)/l₂ = (y - y₂)/m₂ = (z - z₂)/n₂

The shortest distance (S.D.) between L₁ and L₂ is the projection of the line segment joining (x₁, y₁, z₁) and (x₂, y₂, z₂) onto the line of shortest distance.

The direction cosines (l, m, n) of the shortest distance line are proportional to (m₁n₂ - m₂n₁), (n₁l₂ - n₂l₁), (l₁m₂ - l₂m₁).

The formula for the shortest distance is:

S.D. = | (x₂ - x₁)l + (y₂ - y₁)m + (z₂ - z₁)n |

where l, m, n are the actual direction cosines (not ratios) of the S.D. line.

A more direct formula using a determinant is:

S.D. = | Determinant | / sqrt(Σ(m₁n₂ - m₂n₁)² )
where Determinant =

x₂ - x₁	y₂ - y₁	z₂ - z₁
l₁	m₁	n₁
l₂	m₂	n₂

The denominator sqrt(Σ(m₁n₂ - m₂n₁)²) is the magnitude of the cross product of the direction vectors, which is sqrt((m₁n₂ - m₂n₁)² + (n₁l₂ - n₂l₁)² + (l₁m₂ - l₂m₁)²).

If S.D. = 0, the lines are coplanar (either intersecting or parallel).

Equation of the Shortest Distance Line

The line of shortest distance is defined as the intersection of two planes:

1. Plane 1: The plane containing line L₁ and the shortest distance line.
2. Plane 2: The plane containing line L₂ and the shortest distance line.

The equation of Plane 1 is given by the determinant:

x - x₁	y - y₁	z - z��
l₁	m₁	n₁
l	m	n

= 0

The equation of Plane 2 is given by the determinant:

x - x₂	y - y₂	z - z₂
l₂	m₂	n₂
l	m	n

= 0

where (l, m, n) are the direction ratios of the S.D. line, found by l = m₁n₂ - m₂n₁, etc.

The Sphere

General Equation of a Sphere

A sphere is the locus of a point in 3D space that is at a constant distance (the radius) from a fixed point (the center).

• Center-Radius Form:
  If the center is (a, b, c) and the radius is r, the equation is:

  (x - a)² + (y - b)² + (z - c)² = r²

• General Equation:
  Expanding the form above, we get the general equation:

  x² + y² + z² + 2ux + 2vy + 2wz + d = 0

For the general equation:

• Center: (-u, -v, -w)
• Radius: r = sqrt(u² + v² + w² - d)

Sphere through Origin with Intercepts

This is a sphere that passes through the origin (0, 0, 0) and makes intercepts a, b, and c on the x, y, and z axes, respectively.

This means the sphere passes through the four points: (0, 0, 0), (a, 0, 0), (0, b, 0), and (0, 0, c).

Substituting these points into the general equation gives d=0, u=-a/2, v=-b/2, and w=-c/2.

The equation of such a sphere is:

x² + y² + z² - ax - by - cz = 0

Its center is (a/2, b/2, c/2).

Properties and Intersections of Spheres

Section of a Sphere by a Plane

The intersection of a sphere and a plane is always a **circle**.

Let the sphere have center C and radius R. Let the plane be at a perpendicular distance p from C.

• The center of the circular section is the foot of the perpendicular from C to the plane.
• The radius r of the circle is given by Pythagoras' theorem:

  r² = R² - p²

Great Circle

A **great circle** is the largest possible circle that can be drawn on a sphere. It is the section of the sphere by a plane that passes *through the center* of the sphere.

For a great circle, the perpendicular distance p = 0, so its radius r = R (the radius of the sphere).

Sphere Through a Given Circle

A circle in 3D is not defined by a single equation. It is defined by the intersection of two surfaces, usually a sphere and a plane.

Let the circle be defined by:
Sphere: S ≡ x² + y² + z² + 2ux + 2vy + 2wz + d = 0
Plane: P ≡ Lx + My + Nz + Q = 0

The equation of *any* sphere passing through this circle is:

S + kP = 0

where k is a parameter. By changing k, we get different spheres all containing that same circle.

Intersection of Two Spheres

The curve of intersection of two spheres, S₁ = 0 and S₂ = 0, is also a circle.

This circle lies on a plane called the **radical plane**, whose equation is:

P ≡ S₁ - S₂ = 0

(Note: The x², y², z² terms will cancel, leaving a linear equation, which is a plane).

So, the intersection circle is defined by the system: S₁ = 0 and S₁ - S₂ = 0 (or S₂ = 0 and S₁ - S₂ = 0).

Tangent Plane to a Sphere

Tangent Plane at a Point

The equation of the tangent plane to the sphere S = x² + y² + z² + 2ux + 2vy + 2wz + d = 0 at a point (x₁, y₁, z₁) on its surface is given by the T = 0 rule:

The rule involves substitutions:

• x² → x·x₁
• y² → y·y₁
• z² → z·z₁
• 2x → x + x₁
• 2y → y + y₁
• 2z → z + z₁

This gives the equation of the tangent plane:

x·x₁ + y·y₁ + z·z₁ + u(x + x₁) + v(y + y₁) + w(z + z₁) + d = 0

Condition of Tangency for a Plane

A given plane Lx + My + Nz + P = 0 will be a tangent plane (i.e., just touch) a given sphere if and only if:

The perpendicular distance from the center of the sphere to the plane is equal to the radius of the sphere.

Let the sphere be x² + y² + z² + 2ux + 2vy + 2wz + d = 0.
Center C = (-u, -v, -w)
Radius r = sqrt(u² + v² + w² - d)

Perpendicular distance p from C to the plane is:

p = | L(-u) + M(-v) + N(-w) + P | / sqrt(L² + M² + N²)

The **condition of tangency** is:

p = r
( | -Lu - Mv - Nw + P | / sqrt(L² + M² + N²) )² = u² + v² + w² - d

Unit 4: Exam Quick Tips