Score 90+ in FBISE Class 11 Math Unit 3 Vectors – complete solved exercises!
HSA Notes Features
- 100% NBF/FBISE accurate
- All Exercises 3.1–3.4 + Review solved
- Short, exam-ready answers
- Updated 2025-2026 syllabus
- Vector addition, dot/cross products, triple product
- Quick revision notes
- Free PDF with diagrams
- Trusted for top marks
Target Audience
FBISE + Punjab boards. Top schools: Beaconhouse, Roots, LGS, City School, Karachi Grammar, Aitchison, Lawrence, ICG, APS, FPS.
Unit 3 Key Solved Highlights
Exercise 3.1 (Vector Basics, Collinearity, Parallelism)
Q1(i) �PQ = (-7,-5)
Q1(ii) 3�PQ – �RS = (-22,-16)
Q1(iii) 2�PR + 3�PS = (-7,28)
Q1(iv) (-35/2, -13/2)
Q1(v) (-18,155)
Q2(i) Collinear (parallel to x-axis)
Q2(ii) m = -44/7
Q3(i) �x = (2/3, 2/3)
Q3(ii) �u = (-5,8), �v = (7,-11)
Q3(iii) I = (8,-1,-1)
Q4(i) m = 4/3
Q5(i) �v = (1,-1,6); unit = (1/√38, -1/√38, 6/√38)
Q6(i) Right-angled at B
Exercise 3.2 (Dot Product, Projections, Angles)
Q1 �a · �b = -5; �a · �c = 3
Q3(i) �a + �b + �c = (6,0,6); |sum| = 6√2
Q6(i) �AB = (3,2,6), �AC = (5,8,3); �BC = (2,6,-3)
Q7(i) �a + �b + �c = (9,5,9); magnitude √227
Q9(i) |�a + �b|² = |�a|² + |�b|² + 2�a·�b
Exercise 3.3 (Cross Product, Area, Torque)
Q1(i) �a × �b = 4�i -15�j +7�k
Q4(i) Vector magnitude 5 ⊥ to plane: multiple answers possible
Q6(i) �a × �b = 15√15 /4 unit vector check
Q9(ii) |�a| = 2, |�b| = 5
Q14(i) Torque magnitude √57 N·m
Exercise 3.4 (Scalar Triple Product, Coplanarity)
Q1 Cyclic property proved: �a·(�b×�c) = �b·(�c×�a) = �c·(�a×�b)
Q3(i) Coplanar if [�a �b �c] = 0
Q7 Volume = -18 (parallelepiped)
Review Exercise 3
MCQ(i) Direction vector �i + 2�j -2�k; magnitude 3
View PDF for full solutions, diagrams, proofs.
Join classes for 90+. Email: sharjeelmuhammad27@gmail.com
SEO Keywords: class 11 vectors unit 3 solved, fbise nbf math unit 3, exercise 3.1 3.2 3.3 3.4 vectors, federal board vectors full notes 2025, hsa notes unit 3 pdf, 11th class vectors solutions, intermediate math vectors chapter, 90+ marks unit 3, fbise new book vectors 2026, class 11 review exercise 3