FUZY MATH ACADEMY

Class 12 Maths Chapter 11 Three Dimensional Geometry – Exercise 11.5 NCERT Solutions Introduction Exercise 11.5 focuses on the angle between two lines, angle between a line and a plane, and angle between two planes in three‑dimensional geometry. Students learn how to apply dot product and normal vector concepts to calculate these angles. This exercise is crucial for solving advanced problems in 3D geometry. Key Concepts Angle Between Two Lines: If direction ratios of lines are ( a 1 , b 1 , c 1 ) and ( a 2 , b 2 , c 2 ) , then cos ⁡ θ = a 1 a 2 + b 1 b 2 + c 1 c 2 a 1 2 + b 1 2 + c 1 2 ⋅ a 2 2 + b 2 2 + c 2 2 Angle Between Line and Plane: If line has direction ratios ( a , b , c ) and plane has normal vector ( l , m , n ) , then sin ⁡ θ = ∣ a l + b m + c n ∣ a 2 + b 2 + c 2 ⋅ l 2 + m 2 + n 2 Angle Between Two Planes: If normals are ( a 1 , b 1 , c 1 ) and ( a 2 , b 2 , c 2 ) , then cos ⁡ θ = a 1 a 2 + b 1 b 2 + c 1 c 2 a 1 2 + b 1 2 + c 1 2 ⋅ a 2 2 + b 2 2 + c 2 2 Students ...

Class 12 Maths Chapter 11 Three Dimensional Geometry – Exercise 11.4 NCERT Solutions Introduction Exercise 11.4 focuses on the equations of a plane in 3D geometry. Students learn how to represent planes in vector and Cartesian forms, derive equations using point and normal vector, and solve problems involving coplanarity and perpendicularity. This exercise is essential for understanding the geometry of planes in three‑dimensional space. Key Concepts Vector Equation of Plane: Plane through point A ( x 1 , y 1 , z 1 ) and normal vector n ⃗ = ( a , b , c ) : r ⃗ ⋅ n ⃗ = a ⃗ ⋅ n ⃗ where a ⃗ = x 1 i ^ + y 1 j ^ + z 1 k ^ . Cartesian Equation of Plane: a ( x − x 1 ) + b ( y − y 1 ) + c ( z − z 1 ) = 0 or a x + b y + c z + d = 0 Coplanarity Condition: Four points are coplanar if determinant of their position vectors = 0. Perpendicularity Condition: If normals are perpendicular, dot product = 0. Students Frequently Make Mistakes Forgetting to substitute point coordinates correctly. M...

Class 12 Maths Chapter 11 Three Dimensional Geometry – Exercise 11.3 NCERT Solutions Introduction Exercise 11.3 focuses on the equations of a line in space. Students learn how to represent a line in vector form and Cartesian form, and how to solve problems involving points, direction ratios, and direction cosines. This exercise is essential for understanding the geometry of lines in three‑dimensional space. Key Concepts Vector Form of Line: Line through point A ( x 1 , y 1 , z 1 ) and parallel to vector b ⃗ = ( a , b , c ) : r ⃗ = a ⃗ + λ b ⃗ where a ⃗ = x 1 i ^ + y 1 j ^ + z 1 k ^ . Cartesian Form of Line: x − x 1 a = y − y 1 b = z − z 1 c Direction Ratios (DRs): a , b , c are direction ratios of the line. Relation Between Vector and Cartesian Form: Both represent the same line in different notations. Students Frequently Make Mistakes Forgetting to substitute point coordinates correctly. Mixing up vector and Cartesian forms. Errors in identifying direction ratios. Skipping ...

Class 12 Maths Chapter 11 Three Dimensional Geometry – Exercise 11.2 NCERT Solutions Introduction Exercise 11.2 focuses on the direction cosines and direction ratios of a line in 3D geometry. Students learn how to calculate angles made by a line with coordinate axes, find direction cosines, and verify relationships between them. This exercise is essential for understanding equations of lines and planes in three‑dimensional space. Key Concepts Direction Ratios (DRs): If line has vector a ⃗ = ( x , y , z ) , then x , y , z are direction ratios. Direction Cosines (DCs): If line makes angles α , β , γ with x‑, y‑, z‑axes, then l = cos ⁡ α , m = cos ⁡ β , n = cos ⁡ γ Relation Between DCs: l 2 + m 2 + n 2 = 1 Conversion: If DRs are a , b , c , then l = a a 2 + b 2 + c 2 , m = b a 2 + b 2 + c 2 , n = c a 2 + b 2 + c 2 Students Frequently Make Mistakes Forgetting to normalize direction ratios to get cosines. Errors in squaring and summing cosines. Misinterpreting angles with axes. S...

Class 12 Maths Chapter 11 Three Dimensional Geometry – Exercise 11.1 NCERT Solutions Introduction Exercise 11.1 introduces the basic concepts of three‑dimensional coordinate geometry. Students learn how to calculate distance between points, section formula, and midpoint formula in 3D space. This exercise builds the foundation for advanced 3D geometry topics such as equations of lines and planes. Key Concepts Distance Formula in 3D: For points P ( x 1 , y 1 , z 1 ) and Q ( x 2 , y 2 , z 2 ) , P Q = ( x 2 − x 1 ) 2 + ( y 2 − y 1 ) 2 + ( z 2 − z 1 ) 2 Midpoint Formula: Midpoint of P ( x 1 , y 1 , z 1 ) and Q ( x 2 , y 2 , z 2 ) : M = ( x 1 + x 2 2 , y 1 + y 2 2 , z 1 + z 2 2 ) Section Formula: Point dividing line P Q in ratio m : n : ( m x 2 + n x 1 m + n , m y 2 + n y 1 m + n , m z 2 + n z 1 m + n ) Students Frequently Make Mistakes Forgetting to square differences in distance formula. Errors in applying section formula (mixing m and n ). Misplacing signs in coordinates. Ski...