Grade 12
  1. Algebra  1.1. Matrices  1.1.1. Definition and Types of Matrices  1.1.2. Operations on Matrices  1.1.3. Transpose of a matrix  1.1.4. Determinants and their properties  1.1.5. Inverse of a Matrix  1.1.6. Solving Linear Equations Using Matrices  1.1.7. Applications of Matrices  1.2. Complex Numbers  1.2.1. Definition and representation  1.2.2. Algebra of complex numbers  1.2.3. Modulus and argument  1.2.4. Polar and Exponential Forms  1.2.5. De Moivre's Theorem  1.2.6. Roots of complex numbers  1.2.7. Applications in Geometry  1.3. Vectors and 3D Geometry  1.3.1. Vector Algebra  1.3.2. Scalar and Vector Products  1.3.3. Equation of a Line in Space  1.3.4. Equation of a Plane  1.3.5. Distance between lines and planes  1.3.6. Angle between Lines and Planes
  2. Calculus  2.1. Limits and Continuity  2.1.1. Concept of Limits  2.1.2. Continuity of Functions  2.1.3. Types of Discontinuities  2.1.4. Left-hand and Right-hand Limits  2.2. Differentiation  2.2.1. Derivative as a Rate of Change  2.2.2. Techniques of Differentiation  2.2.3. Higher Order Derivatives  2.2.4. Applications of Derivatives  2.2.5. Tangents and Normals  2.2.6. Increasing and Decreasing Functions  2.3. Integration  2.3.1. Indefinite Integrals  2.3.2. Definite Integrals  2.3.3. Integration Techniques  2.3.4. Area under a curve  2.3.5. Applications of Integrals in Geometry  2.4. Differential Equations  2.4.1. Formation of Differential Equations  2.4.2. Order and Degree of Differential Equations  2.4.3. Solutions of Differential Equations  2.4.4. Applications of Differential Equations  2.4.5. Solving First-Order Linear Differential Equations
  3. Probability and Statistics  3.1. Probability  3.1.1. Conditional Probability  3.1.2. Bayes' Theorem  3.1.3. Random Variables  3.1.4. Probability Distributions  3.1.5. Binomial and Normal Distributions  3.2. Statistics  3.2.1. Mean and Standard Deviation  3.2.2. Variance and Covariance  3.2.3. Correlation and Regression  3.2.4. Hypothesis Testing
  4. Linear Programming  4.1. Graphical Method  4.1.1. Feasible and Infeasible Regions  4.1.2. Optimal solutions  4.1.3. Sensitivity Analysis in Graphical Method  4.2. Simplex Method  4.2.1. Formulation of Problems  4.2.2. Solving linear programming problems  4.2.3. Dual Simplex Method
  5. Relations and Functions  5.1. Types of Relations  5.1.1. Reflexive Relations  5.1.2. Symmetric Relations  5.1.3. Transitive Relations  5.1.4. Equivalence Relations  5.2. Types of Functions  5.2.1. Onto Functions  5.2.2. One-to-One Functions  5.2.3. Inverse Functions  5.2.4. Composite Functions  5.3. Inverse Trigonometric Functions  5.3.1. Principal Values in Inverse Trigonometric Functions  5.3.2. Properties and Graphs of Inverse Trigonometric Functions  5.3.3. Applications in Calculus
  6. Advanced Trigonometry  6.1. Trigonometric Equations  6.1.1. General solutions of equations  6.1.2. Transformation formulae  6.2. Hyperbolic Functions  6.2.1. Definitions and properties  6.2.2. Relationships between hyperbolic and trigonometric functions
  7. Mathematical Reasoning  7.1. Logical Reasoning  7.1.1. Statements and Logical Connectives  7.1.2. Tautologies and Contradictions  7.2. Proofs  7.2.1. Direct proof  7.2.2. Indirect Proof  7.2.3. Proof by contradiction  7.2.4. Proof by Mathematical Induction
  8. Applications of Mathematics  8.1. Applications of Calculus  8.1.1. Maxima and Minima Problems  8.1.2. Economics and Biology Applications  8.1.3. Rate of Change in Physics  8.2. Applications of Probability  8.2.1. Risk Analysis  8.2.2. Decision Making  8.2.3. Game Theory

Grade 12 → Algebra → Vectors and 3D Geometry ↓


Distance between lines and planes


In three-dimensional geometry, understanding the concept of distance between lines and planes is essential for solving various mathematical and real-world problems. This topic involves calculating the shortest path between elements such as a line and a plane, which is a fundamental concept in vector and plane geometry.

Distance between a point and a plane

The first basic concept we need to understand is the distance between a point and a plane. Consider a plane defined by the equation:

Ax + By + Cz + D = 0

and a point P(x₁, y₁, z₁). The formula to calculate the distance d from the point P to the plane is given by:

d = |Ax₁ + By₁ + Cz₁ + D| / sqrt(A² + B² + C²)
Example:

Given a plane 2x + 3y - 4z + 5 = 0 and a point P(1, -2, 2), let us find the distance between them. By applying the formula:

d = |2(1) + 3(-2) - 4(2) + 5| / sqrt(2² + 3² + (-4)²)
d = |2 - 6 - 8 + 5| / sqrt(4 + 9 + 16)
d = |-7| / sqrt(29)
d = 7 / sqrt(29)

Distance between a line and a plane

To find the distance between a line and a plane, it is important to establish their position in space. Let's say we have a line represented parametrically:

x = x₀ + at y = y₀ + bt z = z₀ + ct

Here, (x₀, y₀, z₀) is a point on the line and (a, b, c) denotes the direction ratio of the line. Consider the plane with the equation:

Ax + By + Cz + D = 0

The line is parallel to the plane if the dot product of the direction vector (a, b, c) of the line and the normal vector (A, B, C) of the plane is zero:

aA + bB + cC = 0

If the lines are not parallel to the plane, they may intersect, and the possible distance is zero.

Example:

Suppose we have a line with a parametric equation:

x = 2 + 3t y = -1 + 4t z = 5t

And a plane:

3x - y + 2z - 6 = 0

First, we find the direction vector of the line (3, 4, 5) and the normal vector of the plane (3, -1, 2). The dot product is:

3(3) + 4(-1) + 5(2) = 9 - 4 + 10 = 15

Because the dot product is not zero, the line and the plane intersect, that is, the distance is zero.

Distance between parallel line and plane

If a line is parallel to a plane, then the minimum distance between them is measured along the perpendicular drawn from the line to the plane. Given that the direction vector of the line is (a, b, c) and the plane is represented by Ax + By + Cz + D = 0, the distance d is calculated as follows:

d = |Ax₀ + By₀ + Cz₀ + D| / sqrt(A² + B² + C²)

It is important to note that the point (x₀, y₀, z₀) can be any point on the line.

Example:

Suppose we have a line:

x = -1 + ty = 2 + 2t z = 3

And a plane:

4x - 2y + 5z + 1 = 0

Find out if the line is parallel to the plane:

The direction vector of the line is (1, 2, 0) and the normal vector of the plane is (4, -2, 5).

Dot product: 4(1) - 2(2) + 5(0) = 4 - 4 + 0 = 0

Since the dot product is equal to zero, the line is parallel to the plane.

Using (x₀, y₀, z₀) = (-1, 2, 3), the distance from the plane is calculated as:

d = |4(-1) - 2(2) + 5(3) + 1| / sqrt(4² + (-2)² + 5²)
d = |-4 - 4 + 15 + 1| / sqrt(16 + 4 + 25)
d = 8 / sqrt(45)
Simplified, d = 8 / 3sqrt(5)

Visualization example of distance calculation

plane Line The shortest distance

In the visualization above, the black line represents the line in 3D space and the blue rectangle represents the plane. The red dashed line represents the shortest distance, which is perpendicular to both, showing how orthogonal projections determine the minimum distance.

Conclusion

It is necessary to leverage vector algebra and geometric principles to understand the concepts of distance between lines and planes. When calculating distances, determining whether lines are parallel, intersecting, or skew is crucial for applying the proper formulas. By mastering these concepts, one can efficiently solve complex problems in 3D space, which can be used in a variety of fields, including physics, engineering, and computer graphics.


Grade 12 → 1.3.5


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Distance between lines and planes

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