Exercise 14.3 Solutions - Class 11 Mathematics Unit 14 Vectors in Space

Step-by-step solved problems, determinant cross products, and geometric applications for Exercise 14.3, Unit 14: Vectors in Space.Comprehensive solutions covering Vector (Cross) Product, Area of Parallelogram & Triangle, Unit Normal Vectors, and Moment of Force for Punjab Board Class 11 (PCTB).

This page provides complete, step-by-step solved solutions for Exercise 14.3 of Unit 14: Vectors in Space for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All problems on computing cross (vector) products via 3x3 determinants, constructing perpendicular unit normal vectors, verifying vector parallelism (u x v = 0), determining areas of parallelograms and triangles, and evaluating moments of force (torque τ = r x F) are thoroughly resolved for HSSC Part 1 board exam prep.
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Exercise Overview

Exercise 14.3 introduces the Vector (Cross) Product in Unit 14: Vectors in Space in Class 11 Mathematics, defining the non-commutative vector multiplication that generates a vector orthogonal to the plane of two operands according to the Right-Hand Rule.

The exercise systematically explores the algebraic determinant computation of u x v, establishing key properties such as anti-commutativity (u x v = -v x u), parallelism criteria (u x v = 0 for parallel vectors), and derivation of unit normal vectors n̂ = (u x v)/|u x v|. Students apply cross products to spatial geometry by calculating the geometric area of parallelograms (|u x v|) and triangles (1/2 |u x v|), as well as physical engineering mechanics by computing rotational torque via the moment of force vector τ = r x F.

In Punjab Board HSSC Part 1 examinations, Exercise 14.3 is a high-frequency topic in Section B short questions and Section C analytical long questions across Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, and other BISE boards. Mastering the vector cross product is essential for 3D analytical geometry, electromagnetism, and classical rigid body dynamics.

FAQs

Are these Exercise 14.3 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 are the key concepts covered in Exercise 14.3 Class 11 Mathematics?
Exercise 14.3 covers Cross (Vector) Product definitions, 3x3 determinant evaluation, finding unit normal vectors, calculating areas of parallelograms and triangles, and evaluating torque (moment of force).
What is the definition and determinant formula for the Vector (Cross) Product?
The cross product u x v = (|u||v| sin θ) n̂ yields a perpendicular vector whose components are evaluated by expanding the 3x3 matrix determinant with row 1 as unit vectors i, j, k.
How do you find a unit vector perpendicular to two given vectors?
Compute the cross product u x v, then divide by its magnitude: n̂ = ±(u x v) / |u x v| to obtain a unit normal vector perpendicular to both u and v.
How is the cross product used to calculate the area of a parallelogram and a triangle?
The area of a parallelogram with adjacent sides u and v equals |u x v|, while the area of a triangle formed by u and v is half that value: Area = 1/2 |u x v|.
What is the condition for two vectors to be parallel using the cross product?
Two non-zero vectors u and v are parallel (collinear) if and only if their cross product is the zero vector (u x v = 0), since sin(0°) = sin(180°) = 0.
How is the Moment of Force (Torque) calculated using vectors in physics?
The moment of force τ about a pivot point is defined as the cross product of the position vector r and the applied force vector F: τ = r x F.
Are these Class 11 Exercise 14.3 notes completely free to access?
Yes, all solved exercise notes, cross product derivations, and mechanics solutions on MaryamNotes.pk are 100% free and open-access for students and teachers across Pakistan.