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Foundational Topic

Three-Dimensional Geometry

Mathematics Unit 3
25 min read
IAT Core Framework
High Yield

1. Core Concept

Three-dimensional geometry uses algebra to describe shapes, lines, and planes in space by adding a z-axis. It heavily relies on vectors to define directions. A line is defined by the direction it runs parallel to, while a plane is defined by the direction it is perpendicular to (its normal).

2. Key Formulas

Direction Cosines (DCs) & Ratios (DRs)

DCs: l = cos(α), m = cos(β), n = cos(γ) &implies; l2 + m2 + n2 = 1.

DRs (a, b, c) are proportional to DCs:
l = ±a / √(a2 + b2 + c2)

Vector: r = a + λb
Cartesian: (x - x1)/a = (y - y1)/b = (z - z1)/c = λ
Equation of a Line: Passing through point a (or x1, y1, z1), parallel to direction vector b (or DRs a, b, c).
Vector: (r - a) · n = 0 &implies; r · n = a · n = d
Cartesian: A(x - x1) + B(y - y1) + C(z - z1) = 0
Equation of a Plane: Passing through a, normal to vector n (with DRs A, B, C).

Shortest Distance

Between Skew Lines: r = a1 + λb1 and r = a2 + μb2

d = | (a2 - a1) · (b1 × b2) | / | b1 × b2 |

(If lines are parallel (b1 = b2 = b), then d = |(a2 - a1) × b| / |b|)

Distance of a Point from a Plane

Distance of point (x1, y1, z1) from plane Ax + By + Cz + D = 0:

d = |Ax1 + By1 + Cz1 + D| / √(A2 + B2 + C2)

3. Important Results

  • Angle between Two Lines: cos(θ) = (b1 · b2) / (|b1| |b2|)
  • Angle between Two Planes: cos(θ) = (n1 · n2) / (|n1| |n2|)
  • Angle between a Line and a Plane: sin(θ) = (b · n) / (|b| |n|)
    (Notice the sine instead of cosine!)

Condition for Coplanarity of Two Lines

The shortest distance must be zero: (a2 - a1) · (b1 × b2) = 0 (Scalar Triple Product is zero).

In determinant form:

x2-x1    y2-y1    z2-z1
a1          b1          c1
a2          b2          c2

= 0

4. Conceptual Insights

  • Skew Lines: In 2D, lines either intersect or are parallel. In 3D, there is a third option: Skew lines. They do not intersect and are not parallel (imagine a bridge over a river; they cross in space but never touch).
  • Normal to a Plane: The coefficients of x, y, and z in the general equation of a plane (Ax + By + Cz + D = 0) are exactly the Direction Ratios (DRs) of its normal vector (Aiˆ + Bjˆ + Ckˆ).
  • General Point on a Line: By equating the Cartesian equation to λ, any point on the line can be written as (aλ + x1, bλ + y1, cλ + z1). This is your ultimate weapon for intersection problems.

5. Common Mistakes

  • The Cartesian Format Trap: To extract the correct DRs from a line equation, the coefficients of x, y, and z must be exactly +1.
    Wrong: From (2x - 1)/3 = ..., taking DR as 3.
    Right: Rewrite as (x - 1/2) / (3/2) = ..., the actual DR is 3/2.
  • Line-Plane Angle Mix-up: Students often blindly use the dot product formula and solve for cos(θ). The angle between a line (b) and a plane's normal (n) is (90° - θ). Therefore, use sin(θ).
  • Forgetting "D": In the plane equation r · n = d, 'd' is only the perpendicular distance from the origin if n is a unit vector (nˆ). If it's a general vector, divide by |n| first.

6. Example Applications

Family of Planes

The equation of a plane passing through the line of intersection of two planes P1 = 0 and P2 = 0 is given by P1 + λP2 = 0. Use the given extra condition (like passing through a specific point) to find λ.

(h - x1)/A = (k - y1)/B = (l - z1)/C = -2(Ax1 + By1 + Cz1 + D) / (A2 + B2 + C2)
Foot of Perpendicular / Image: To find the image (h, k, l) of point (x1, y1, z1) across plane Ax + By + Cz + D = 0.
(For just the foot of the perpendicular, drop the 2).

7. IAT Exam Focus Points

  • Shortest Distance Calculation: This is the highest yield topic. If you see two lines in vector format, immediately calculate (b1 × b2) and apply the formula.
  • Coplanarity Determinant: You will often be given two lines with an unknown variable 'k' and told they are coplanar. Set the 3×3 determinant equal to 0 and solve for k.
  • Vector-Cartesian Fluency: You must be able to convert between r = (iˆ + 2jˆ) + λ(3kˆ) and its Cartesian counterpart in under 10 seconds.
  • Intersection of Line and Plane: Parametrize the line in terms of λ, plug those (x, y, z) coordinates into the plane's equation, solve for λ, and plug it back in to find the point.

8. Practice Mock Test

Ready to test your knowledge?

Take a quick 15-question assessment specifically designed for Three-Dimensional Geometry. Challenge yourself with IAT-level questions.

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End of Chapter

Three-Dimensional Geometry

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