Exercise 14.2 Solutions - Class 11 Mathematics Unit 14 Vectors in Space

Step-by-step solved problems, geometric proofs, and physics applications for Exercise 14.2, Unit 14: Vectors in Space.Comprehensive solutions covering Scalar (Dot) Product, Vector Projections, Orthogonality, Angle Between Vectors, and Work Done for Punjab Board Class 11 (PCTB).

This page provides complete, step-by-step solved solutions for Exercise 14.2 of Unit 14: Vectors in Space for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All problems on computing scalar (dot) products via components (u · v = u₁v₁ + u₂v₂ + u₃v₃), determining interior angles between 3D vectors, testing vector orthogonality (u · v = 0), evaluating scalar and vector projections, and calculating physical work done by constant force vectors (W = F · d) are thoroughly solved for HSSC Part 1 exam success.
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Exercise Overview

Exercise 14.2 introduces the Scalar (Dot) Product in Unit 14: Vectors in Space in Class 11 Mathematics, establishing the algebraic and geometric foundations for measuring vector alignment, angular separation, and directional projections in 3D space.

The exercise thoroughly develops both theoretical definitions and analytical computations: applying the geometric formula u · v = |u||v| cos θ alongside the component form u₁v₁ + u₂v₂ + u₃v₃; deriving the orthogonality criterion (u · v = 0 for mutually perpendicular vectors); finding the precise scalar projection (u · v̂) and vector projection of one vector along another; and solving classical mechanical physics applications such as calculating total work done by concurrent forces W = F · d.

In Punjab Board HSSC Part 1 examinations, Exercise 14.2 is a core topic, consistently featured in Section B short conceptual calculations (finding unknown parameters for perpendicular vectors) and Section C long application problems across Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, and other BISE boards. Mastering the dot product is essential for subsequent cross products, vector geometry, and physical mechanics.

FAQs

Are these Exercise 14.2 solutions according to the Punjab Board syllabus?
Yes, these solutions strictly follow the official Punjab Curriculum and Textbook Board (PCTB / PECTAA) Class 11 Mathematics syllabus for Unit 14 (Vectors in Space) for the 2026-2027 academic session.
What core topics are covered in Exercise 14.2 Class 11 Mathematics?
Exercise 14.2 covers Dot (Scalar) Product definitions and component forms, calculating angles between vectors, testing orthogonality, computing scalar/vector projections, and calculating work done by a constant force.
What is the definition and component formula for the Dot (Scalar) Product?
Geometrically, u · v = |u| |v| cos θ. In Cartesian components where u = u₁i + u₂j + u₃k and v = v₁i + v₂j + v₃k, the dot product is u · v = u₁v₁ + u₂v₂ + u₃v₃.
How do you find the angle between two 3D vectors?
The angle θ is found using cos θ = (u · v) / (|u| |v|) = (u₁v₁ + u₂v₂ + u₃v₃) / [√(u₁² + u₂² + u₃²) √(v₁² + v₂² + v₃²)], where 0 ≤ θ ≤ π.
When are two vectors orthogonal (perpendicular) to each other?
Two non-zero vectors u and v are orthogonal if and only if their dot product equals zero (u · v = 0), since cos(90°) = 0.
What is the formula for the projection of vector u along vector v?
The scalar projection of u on v is given by (u · v) / |v| = u · v̂, while the vector projection is given by [(u · v) / |v|²] v.
How is the dot product used to calculate work done by a constant force?
Work done is defined as the scalar product of force vector F and displacement vector d: W = F · d = |F| |d| cos θ, measured in Joules.
Are these Class 11 Exercise 14.2 notes completely free to access?
Yes, all solved exercise notes, vector mechanics solutions, and board exam guides on MaryamNotes.pk are 100% free and open-access for students and teachers across Pakistan.