Exercise 14.1 Solutions - Class 11 Mathematics Unit 14 Vectors in Space

Step-by-step solved problems, 3D coordinate geometry, and vector proofs for Exercise 14.1, Unit 14: Vectors in Space.Comprehensive solutions covering Vector Addition, Magnitudes, Distance Formula in Space, and Direction Cosines for Punjab Board Class 11 (PCTB).

This page provides complete, step-by-step solved solutions for Exercise 14.1 of Unit 14: Vectors in Space for Class 11 Mathematics under the Punjab Board (PCTB / PECTAA). All problems on 3D Cartesian coordinates, vector arithmetic (addition, scalar multiplication, unit vectors), evaluating 3D Euclidean distances between points, and finding direction angles and direction cosines satisfying cos²α + cos²β + cos²γ = 1 are thoroughly resolved for HSSC Part 1 exam success.
Back
Exercise 13.3
Next
Exercise 14.2
Advertisement

Exercise Overview

Exercise 14.1 opens Unit 14: Vectors in Space in Class 11 Mathematics, extending 2D plane vector algebra into the three-dimensional Cartesian coordinate system (x, y, z) with basis unit vectors (i, j, k).

The exercise systematically explores 3D spatial geometry: defining position vectors r = xi + yj + zk; computing Euclidean distance and displacement vectors between coordinates A(x₁, y₁, z₁) and B(x₂, y₂, z₂); calculating vector lengths |v| = √(x² + y² + z²) and unit vectors; and analyzing spatial orientation via direction angles (α, β, γ) and direction cosines (cos α, cos β, cos γ), proving the foundational geometric identity cos²α + cos²β + cos²γ = 1.

In Punjab Board HSSC Part 1 examinations, Exercise 14.1 questions are frequently featured in Section A MCQs, Section B short conceptual calculations, and Section C vector proofs across Lahore, Rawalpindi, Gujranwala, Faisalabad, Multan, and other BISE boards. Mastering these spatial vector representations is essential for scalar/vector products, 3D analytic geometry, and classical mechanics in physics.

FAQs

Are these Exercise 14.1 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 14.1 Class 11 Mathematics?
Exercise 14.1 covers 3D rectangular coordinates, vector operations (addition, scalar multiplication), calculating 3D magnitudes, finding vectors between two points, and computing direction cosines.
How do you find the magnitude of a vector in 3D space?
For a vector v = xi + yj + zk, its magnitude is given by |v| = √(x² + y² + z²), derived from the 3D Pythagorean theorem.
How do you compute the vector joining two points A(x₁, y₁, z₁) and B(x₂, y₂, z₂)?
The displacement vector is given by AB = (x₂ - x₁)i + (y₂ - y₁)j + (z₂ - z₁)k, and the distance between them is the magnitude |AB| = √[(x₂ - x₁)² + (y₂ - y₁)² + (z₂ - z₁)²].
What are Direction Angles and Direction Cosines in 3D vector geometry?
Direction angles (α, β, γ) are angles made by a vector with positive x, y, and z axes. Their cosines (cos α = x/|r|, cos β = y/|r|, cos γ = z/|r|) are called direction cosines.
Why is cos²α + cos²β + cos²γ = 1 for any space vector?
Substituting cos α = x/|r|, cos β = y/|r|, and cos γ = z/|r| yields (x² + y² + z²)/|r|² = |r|²/|r|² = 1, proving that the sum of the squares of direction cosines is always unity.
Can students read these Exercise 14.1 solutions online?
Yes, students can read all Exercise 14.1 3D vector calculations and direction cosine proofs directly online on MaryamNotes.pk on any web browser without downloading.
Are these Class 11 Exercise 14.1 notes completely free to access?
Yes, all solved exercise notes, 3D geometry diagrams, and vector study materials on MaryamNotes.pk are 100% free and open-access for students and teachers across Pakistan.