Exercise 14.4 Solutions - Class 11 Mathematics Unit 14 Vectors in Space

Step-by-step solved problems, 3D volume calculations, and vector proofs for Exercise 14.4, Unit 14: Vectors in Space.Comprehensive solutions covering Scalar Triple Product, Volume of Parallelepiped & Tetrahedron, Coplanarity of Vectors, and Real-World Applications for Punjab Board Class 11 (PCTB).

This page provides complete, step-by-step solved solutions for Exercise 14.4 of Unit 14: Vectors in Space for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All problems on evaluating the Scalar Triple Product u · (v x w) via 3x3 determinants, calculating 3D volumes of parallelepipeds (V = |[u v w]|) and tetrahedra (V = 1/6 |[u v w]|), proving coplanarity of vectors and points ([u v w] = 0), and solving real-world vector applications are thoroughly resolved for HSSC Part 1 exam success.
Back
Exercise 14.3
Advertisement

Exercise Overview

Exercise 14.4 concludes Unit 14: Vectors in Space and completes the Class 11 Mathematics curriculum by mastering the Scalar Triple Product (Box Product), unifying 3D spatial geometry with matrix determinant calculus.

The exercise systematically develops the algebraic evaluation of the scalar triple product u · (v x w) = [u v w] using 3x3 component determinants: proving cyclic permutations, dot-cross exchange rules, and zero conditions. Students apply these concepts to calculate exact geometric volumes of 3D solids-determining the volume of parallelepipeds V = |[u v w]| and tetrahedra V = 1/6 |[u v w]| formed by coterminous edge vectors-and establishing the fundamental coplanarity condition [u v w] = 0 to test whether multiple vectors or four spatial points reside on a common plane.

In Punjab Board HSSC Part 1 examinations, Exercise 14.4 is a hallmark section for high-yield Section B conceptual short questions and Section C long analytical proofs across Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, and other BISE boards. Mastering the scalar triple product equips students with vital mathematical competence for multivariable calculus, linear algebra, engineering mechanics, and computer graphics.

FAQs

Are these Exercise 14.4 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 14.4 Class 11 Mathematics?
Exercise 14.4 covers the Scalar Triple Product [u v w], its 3x3 determinant evaluation, calculating the volume of a parallelepiped and tetrahedron, and proving coplanarity of vectors and points in space.
What is the Scalar Triple Product and how is it evaluated using determinants?
The Scalar Triple Product u · (v x w) represents a scalar value computed by evaluating the 3x3 determinant whose rows contain the Cartesian components of vectors u, v, and w.
What are the main algebraic properties of the Scalar Triple Product?
Key properties include cyclic invariance [u v w] = [v w u] = [w u v], interchangeability of dot and cross u · (v x w) = (u x v) · w, and that [u v w] = 0 if any two vectors are identical or parallel.
How do you find the volume of a parallelepiped and a tetrahedron using vectors?
The volume of a parallelepiped with coterminous edges u, v, w is V = |[u v w]|. The volume of a tetrahedron formed by the same coterminous edges is V = 1/6 |[u v w]|.
What is the condition for three 3D vectors to be coplanar?
Three non-zero vectors u, v, and w are coplanar (lie in the same plane) if and only if their scalar triple product is zero: [u v w] = 0, meaning the volume of the spanned parallelepiped is zero.
How do you test if four given points in space lie on the same plane?
Form three displacement vectors sharing a common initial point, such as AB, AC, and AD. If their scalar triple product [AB AC AD] = 0, then points A, B, C, and D are coplanar.
Are these Class 11 Exercise 14.4 notes completely free to access?
Yes, all solved exercise notes, 3D volume calculations, and vector geometry proofs on MaryamNotes.pk are 100% free and open-access for students and educators across Pakistan.