Distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) is √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
Distance between a point (x₁, y₁, z₁) and a line (x - x₀)/a = (y - y₀)/b = (z - z₀)/c is |(x₁ - x₀)a + (y₁ - y₀)b + (z₁ - z₀)c| / √(a² + b² + c²)
Equation of a plane ∈ normal form is x/a + y/b + z/c = 1, where a, b, c are direction ratios of normal to the plane
Equation of a line ∈ symmetric form is x/x₁ = y/y₁ = z/z₁
Direction cosines of a line are l = cosα, m = cosβ, n = cosγ, where α, β, γ are angles with x, y, z axes respectively
l² + m² + n² = 1
Equation of a sphere with center (x₀, y₀, z₀) and radius r is (x - x₀)² + (y - y₀)² + (z - z₀)² = r²
Equation of a plane passing through a point (x₁, y₁, z₁) and having direction ratios a, b, c is a(x - x₁) + b(y - y₁) + c(z - z₁) = 0

🪤 The 5 Mistakes That Cost Marks

Not using the correct formula for distance between two points ∈ 3D space
Forgetting to consider the direction ratios of the normal to the plane while writing the equation of the plane
Not using the symmetric form of the equation of a line while finding the direction cosines
Not checking if the direction cosines satisfy the condition l² + m² + n² = 1
Not using the correct equation of a sphere while finding the center and radius of the sphere

✏️ 3 Solved PYQs

Find the distance between the points (1, 2, 3) and (4, 5, 6)
- Using the formula, distance = √((4 - 1)² + (5 - 2)² + (6 - 3)²) = √(3² + 3² + 3²) = √(9 + 9 + 9) = √27 = 3√3
Find the equation of the plane passing through the point (1, 2, 3) and having direction ratios 1, 2, 3
- Using the formula, equation of plane is 1(x - 1) + 2(y - 2) + 3(z - 3) = 0
- Simplifying, x + 2y + 3z - 1 - 4 - 9 = 0, x + 2y + 3z - 14 = 0
Find the direction cosines of the line passing through the points (1, 2, 3) and (4, 5, 6)
- Using the formula, direction ratios are 4 - 1, 5 - 2, 6 - 3, i.e., 3, 3, 3
- Direction cosines are l = 3/√(3² + 3² + 3²) = 3/√27 = 1/√3, m = 3/√27 = 1/√3, n = 3/√27 = 1/√3

🧠 The One Thing Most Students Get Wrong

Most students get confused between the equation of a line ∈ symmetric form and the equation of a plane ∈ normal form
They forget to consider the direction ratios of the normal to the plane while writing the equation of the plane
They also forget to check if the direction cosines satisfy the condition l² + m² + n² = 1

👁️ Ayush's Note

To solve 3D geometry problems, first visualize the problem and try to identify the key elements such as points, lines, and planes
Use the correct formulas and equations to find the required quantities
Always check your calculations and ensure that the direction cosines satisfy the condition l² + m² + n² = 1
Practice is key to mastering 3D geometry, so make sure to practice a variety of problems

🔁 Last 5 Minutes Box

Revision of key formulas and equations
Practice of quick calculations and estimations
Focus on common mistakes and how to avoid them
Quick review of solved problems and examples
Deep breathing exercises to calm the mind and focus

📝 Practice MCQs

1. What is the distance between the points (1, 2, 3) and (4, 5, 6)?

A) 2√3

B) 3√3

C) 4√3

D) 5√3

Answer: B) 3√3.

2. What is the equation of the plane passing through the point (1, 2, 3) and having direction ratios 1, 2, 3?

A) x + 2y + 3z - 14 = 0

B) x + 2y + 3z + 14 = 0

C) x - 2y - 3z + 14 = 0

D) x - 2y - 3z - 14 = 0

Answer: A) x + 2y + 3z - 14 = 0.

3. What are the direction cosines of the line passing through the points (1, 2, 3) and (4, 5, 6)?

A) 1/√3, 1/√3, 1/√3

B) 1/√2, 1/√2, 0

C) 1, 0, 0

D) 0, 1, 0

Answer: A) 1/√3, 1/√3, 1/√3.

4. What is the equation of the sphere with center (1, 2, 3) and radius 4?

A) (x - 1)² + (y - 2)² + (z - 3)² = 16

B) (x - 1)² + (y - 2)² + (z - 3)² = 4

C) (x + 1)² + (y + 2)² + (z + 3)² = 16

D) (x + 1)² + (y + 2)² + (z + 3)² = 4

Answer: A) (x - 1)² + (y - 2)² + (z - 3)² = 16.

5. What is the distance between the point (1, 2, 3) and the line (x - 1)/1 = (y - 2)/2 = (z - 3)/3?

A) 0

B) 1

C) 2

D) 3

Answer: A) 0.

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This post was curated by Jules, Exam Compass Bot, and edited for accuracy by Ayush.

📚 Related Topics

Continue your revision with these related guides:

Distance between two points (x₁, y₁, z₁) and (x₂, y₂, z₂) is √((x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²)
Distance between a point (x₁, y₁, z₁) and a line (x - x₀)/a = (y - y₀)/b = (z - z₀)/c is |(x₁ - x₀)a + (y₁ - y₀)b + (z₁ - z₀)c| / √(a² + b² + c²)
Equation of a plane ∈ normal form is x/a + y/b + z/c = 1, where a, b, c are direction ratios of normal to the plane
Equation of a line ∈ symmetric form is x/x₁ = y/y₁ = z/z₁
Direction cosines of a line are l = cosα, m = cosβ, n = cosγ, where α, β, γ are angles with x, y, z axes respectively
l² + m² + n² = 1
Equation of a sphere with center (x₀, y₀, z₀) and radius r is (x - x₀)² + (y - y₀)² + (z - z₀)² = r²
Equation of a plane passing through a point (x₁, y₁, z₁) and having direction ratios a, b, c is a(x - x₁) + b(y - y₁) + c(z - z₁) = 0