Class 9 Mathematics Chapter 10 Vector Algebra: Important Questions with Complete Solutions

Vector Algebra has evolved into a high-weight topic in modern CBSE assessments. The 2024-25 rationalized syllabus emphasizes conceptual understanding over rote memorization, meaning examiners now focus on application-based questions rather than simple definitions. In the 2026-27 board pattern, expect: • **1-mark MCQs** testing recognition of vector properties (e.g., identifying whether two vectors are parallel or perpendicular) • **2-3 mark short-answer questions** requiring calculation of dot products, cross products, or magnitude • **5-mark long-answer questions** combining multiple concepts, such as proving geometric theorems using vectors or solving real-world physics applications • **HOTS questions** that demand logical reasoning—for instance, determining when two forces (vectors) result in zero net acceleration The shift toward integrated physics-mathematics problems means you must be fluent in both notation and geometric interpretation. Questions no longer ask 'Define a vector' but instead ask 'Find the angle between two velocity vectors given their components.' This document provides the exact question types and difficulty gradients your board examiner will use. Master these patterns, and you'll handle any variant thrown at you.

**Q1.** A quantity that has only magnitude and no direction is called: (A) Vector (B) Scalar (C) Unit vector (D) Position vector **Answer: (B) Scalar** Explanation: Scalars like temperature, mass, and speed possess magnitude only. Vectors require both magnitude and direction (e.g., velocity, displacement, force). --- **Q2.** The dot product of two perpendicular vectors is: (A) Zero (B) Maximum (C) Minimum (D) Equal to product of magnitudes **Answer: (A) Zero** Explanation: For vectors **u** and **v**, dot product = |**u**| |**v**| cos θ. When θ = 90°, cos 90° = 0, so **u** · **v** = 0. --- **Q3.** If |**a**| = 3 and |**b**| = 4, and **a** is parallel to **b**, then **a** · **b** equals: (A) 12 (B) −12 (C) 7 (D) Either 12 or −12 **Answer: (D) Either 12 or −12** Explanation: If **a** and **b** are parallel, θ = 0° or 180°. At 0°, **a** · **b** = 3 × 4 × 1 = 12. At 180°, **a** · **b** = 3 × 4 × (−1) = −12. --- **Q4.** The cross product **a** × **b** gives: (A) A scalar quantity (B) A vector perpendicular to both **a** and **b** (C) A vector parallel to **a** (D) Zero **Answer: (B) A vector perpendicular to both **a** and **b**** Explanation: Cross product yields a vector whose direction is given by the right-hand rule and is perpendicular to the plane containing **a** and **b**. --- **Q5.** If **a** = 2**i** + 3**j** and **b** = **i** − 2**j**, then **a** · **b** =: (A) 5 (B) −4 (C) 4 (D) 6 **Answer: (B) −4** Explanation: **a** · **b** = (2)(1) + (3)(−2) = 2 − 6 = −4

**Q1.** Define unit vector. If **a** = 3**i** + 4**j**, find the unit vector along **a**. **Answer:** A unit vector is a vector with magnitude 1 in a given direction. Given **a** = 3**i** + 4**j**, |**a**| = √(3² + 4²) = √25 = 5 Unit vector = **a**/|**a**| = (3**i** + 4**j**)/5 = (3/5)**i** + (4/5)**j** --- **Q2.** If **p** = 2**i** + **j** − **k** and **q** = **i** − 3**j** + 2**k**, find **p** · **q**. **Answer:** **p** · **q** = (2)(1) + (1)(−3) + (−1)(2) = 2 − 3 − 2 = −3 --- **Q3.** Two vectors **m** and **n** have magnitudes 6 and 8 respectively. If the angle between them is 60°, find **m** · **n**. **Answer:** **m** · **n** = |**m**| |**n**| cos θ = 6 × 8 × cos 60° = 48 × (1/2) = 24 --- **Q4.** State the difference between scalar product (dot product) and vector product (cross product) with one example each. **Answer:** Scalar Product: Results in a scalar; formula = |**a**| |**b**| cos θ; Example: Work = Force · Displacement Vector Product: Results in a vector; magnitude = |**a**| |**b**| sin θ; direction perpendicular to both; Example: Torque = Force × Position vector --- **Q5.** If **r** = **i** + 2**j** + 3**k** and **s** = 2**i** − **j** + **k**, calculate the magnitude of **r** − **s**. **Answer:** **r** − **s** = (1−2)**i** + (2−(−1))**j** + (3−1)**k** = −**i** + 3**j** + 2**k** |**r** − **s**| = √((−1)² + 3² + 2²) = √(1 + 9 + 4) = √14

**Q1.** Find the angle between vectors **a** = 2**i** + **j** − 2**k** and **b** = **i** + **j** + **k**. **Answer:** **a** · **b** = (2)(1) + (1)(1) + (−2)(1) = 2 + 1 − 2 = 1 |**a**| = √(4 + 1 + 4) = √9 = 3 |**b**| = √(1 + 1 + 1) = √3 cos θ = (**a** · **b**)/(|**a**| |**b**|) = 1/(3√3) = 1/(3√3) × (√3/√3) = √3/9 θ = cos⁻¹(√3/9) ≈ 73.9° --- **Q2.** Prove that the vectors **u** = **i** + 2**j** − **k** and **v** = −**i** − 2**j** + **k** are parallel. **Answer:** Two vectors are parallel if one is a scalar multiple of the other, i.e., **v** = k**u**. **v** = −**i** − 2**j** + **k** = −(1)**i** − (1)2**j** + (−1)(−**k**) = −1(**i** + 2**j** − **k**) = −1 × **u** Since **v** = −**u**, the vectors are parallel (in opposite directions). --- **Q3.** If **c** = 3**i** and **d** = 4**j**, verify that |**c** + **d**|² = |**c**|² + |**d**|² when **c** ⊥ **d**. **Answer:** **c** + **d** = 3**i** + 4**j** |**c** + **d**|² = (3)² + (4)² = 9 + 16 = 25 |**c**|² = (3)² = 9 |**d**|² = (4)² = 16 |**c**|² + |**d**|² = 9 + 16 = 25 ∴ |**c** + **d**|² = |**c**|² + |**d**|² ✓ (Verified; perpendicular vectors satisfy Pythagoras' theorem) --- **Q4.** Two forces **F₁** = 5**i** + 3**j** and **F₂** = 2**i** − 4**j** act on a particle. Find the resultant force and its magnitude. **Answer:** Resultant **R** = **F₁** + **F₂** = (5+2)**i** + (3−4)**j** = 7**i** − **j** Magnitude |**R**| = √(7² + (−1)²) = √(49 + 1) = √50 = 5√2 units

**Q1.** Two vectors **p** and **q** are given by **p** = 2**i** − 3**j** + **k** and **q** = **i** + 2**j** − 2**k**. Find: (i) **p** · **q**, (ii) |**p**|, (iii) |**q**|, (iv) the angle between **p** and **q**, and (v) verify the cosine formula. **Complete Solution:** (i) **p** · **q** = (2)(1) + (−3)(2) + (1)(−2) = 2 − 6 − 2 = −6 (ii) |**p**| = √(2² + (−3)² + 1²) = √(4 + 9 + 1) = √14 (iii) |**q**| = √(1² + 2² + (−2)²) = √(1 + 4 + 4) = √9 = 3 (iv) cos θ = (**p** · **q**)/(|**p**| × |**q**|) = −6/(√14 × 3) = −6/(3√14) = −2/√14 = −2√14/14 = −√14/7 θ = cos⁻¹(−√14/7) ≈ 131.81° (v) Verification: Since cos θ = −√14/7 < 0, the angle is obtuse (between 90° and 180°). This confirms that **p** · **q** is negative, which is consistent with an obtuse angle. --- **Q2.** A particle moves from point A(1, 2, 3) to point B(4, 5, 6). A force **F** = 2**i** + **j** − **k** acts on it. Calculate the work done by the force. (Hint: Work = **F** · displacement) **Complete Solution:** Displacement vector **s** = B − A = (4−1)**i** + (5−2)**j** + (6−3)**k** = 3**i** + 3**j** + 3**k** Force **F** = 2**i** + **j** − **k** Work = **F** · **s** = (2)(3) + (1)(3) + (−1)(3) = 6 + 3 − 3 = 6 Joules --- **Q3.** Show that the vectors **a** = **i** − 2**j** + **k**, **b** = 2**i** + **j** − **k**, and **c** = **i** + 3**j** − 2**k** form a triangle. Find the angles of the triangle. **Complete Solution:** For these to form a triangle sides must satisfy: **a** + **b** + **c** = 0 OR they represent the three sides. Let's check if they're coplanar and form sides: Let side 1 = **a**, side 2 = **b**, side 3 = **c** Side opposite to **a**: **b** − **c** = (2−1)**i** + (1−3)**j** + (−1−(−2))**k** = **i** − 2**j** + **k** = **a** ✓ Now, angle at A (between sides **b** and **c**): cos A = (**b** · **c**)/(|**b**| |**c**|) **b** · **c** = (2)(1) + (1)(3) + (−1)(−2) = 2 + 3 + 2 = 7 |**b**| = √(4 + 1 + 1) = √6 |**c**| = √(1 + 9 + 4) = √14 cos A = 7/(√6 × √14) = 7/√84 = 7/(2√21) = 7√21/42 = √21/6 A ≈ 23.4° Angle at B (between sides **a** and −**c**): **a** · (−**c**) = −7 cos B = −7/(√6 × √14) = −√21/6 B ≈ 156.6° (Note: Angle verification: A + B + C = 180° can be verified numerically.)

**Case Study: Navigation and Force Analysis** A drone delivery service calculates the resultant velocity of a delivery package in three-dimensional airspace. The wind applies a velocity vector **w** = 2**i** − **j** + **k** m/s, and the drone's velocity relative to the air is **d** = 5**i** + 3**j** − 2**k** m/s. Engineers must determine: **(a)** The actual velocity of the drone relative to the ground (resultant). **(b)** The speed (magnitude) of the drone's motion. **(c)** The angle between the wind direction and drone direction to assess stability. **(d)** Whether the wind and drone motion are perpendicular (which would cause minimum interference). **Solution:** **(a)** Resultant velocity **v** = **d** + **w** = (5+2)**i** + (3−1)**j** + (−2+1)**k** = 7**i** + 2**j** − **k** m/s **(b)** Speed |**v**| = √(7² + 2² + (−1)²) = √(49 + 4 + 1) = √54 = 3√6 ≈ 7.35 m/s **(c)** **w** · **d** = (2)(5) + (−1)(3) + (1)(−2) = 10 − 3 − 2 = 5 |**w**| = √(4 + 1 + 1) = √6 |**d**| = √(25 + 9 + 4) = √38 cos θ = 5/(√6 × √38) = 5/√228 = 5/(2√57) ≈ 0.33 θ ≈ 70.5° **(d)** Since **w** · **d** = 5 ≠ 0, the vectors are NOT perpendicular. They form a 70.5° angle, suggesting moderate interaction between wind and drone motion. **Key Learning:** Vector operations allow engineers to decompose complex 3D motion into components, predict resultant behavior, and optimize navigation paths—a real-world application of Chapter 10 concepts.

At CBSETUTOR.ai, our adaptive AI tutor is specifically trained on the 2024-25 CBSE Class 9 rationalized syllabus and the evolving 2026-27 board examination patterns. Here's how we ensure mastery of Vector Algebra Chapter 10: **Personalized Diagnostic Tests:** On day one, the AI assesses your current level—whether you confuse scalars with vectors or already struggle with cross products. It then generates a custom learning path. **Daily Drill Sets:** Each session serves 3–5 questions matching YOUR weakness zones. If you score 60% on dot products but 90% on definitions, tomorrow's drills focus on dot product applications, not repetition. **Worked Example Playback:** For every question you attempt, if incorrect, the AI walks you through step-by-step solutions with visual vector diagrams, explaining the 'why' behind each operation, not just the answer. **Mixed-Difficulty Progression:** We don't keep you on easy MCQs. The AI automatically escalates: 1-mark MCQs → 2-mark short-answer → 3-mark problems → 5-mark integrated questions. You're always at your edge, building confidence without plateauing. **Board-Pattern Simulation:** Every Friday, take a full-length mock test replicating the exact 2026-27 board format: 10 MCQs, 8 short-answer, 6 long-answer, 1 HOTS. Instant feedback on time management and accuracy. **Real-Time Concept Reinforcement:** When you make an error—say, forgetting that cos 90° = 0 in dot products—the AI immediately reinforces that concept across 5 different question contexts before moving forward. **Teachers' Analytics Dashboard:** If you're a parent or school educator, track progress weekly: accuracy %, time per question, concept mastery score. Identify which students need Vector Algebra intervention early. Start a 3-day free trial at cbsetutor.ai to experience this structured, evidence-based approach to mastering Class 9 Mathematics Chapter 10.

Use this checklist in the 2 weeks before your board exam to ensure you've covered every angle of Vector Algebra Chapter 10: ✓ **Conceptual Clarity:** - Can you define scalars, vectors, and unit vectors without referring to notes? - Do you know the geometric meaning of dot product (angle measurement) and cross product (area/perpendicularity)? - Can you explain why **a** · **b** = 0 implies perpendicularity? ✓ **Calculation Speed:** - Solve 5 dot product problems in <10 minutes without error. - Calculate magnitudes of 3D vectors in <5 minutes each. - Find angles between two vectors in <8 minutes. ✓ **Application Problems:** - Solve at least 2 work/force problems (physics integration). - Attempt 1 navigation/displacement problem (real-world context). - Practice 1 proof-based problem (e.g., proving parallel vectors). ✓ **Pattern Recognition:** - Solve all 5 MCQs under exam pressure (no notes). - Complete all 5 two-mark questions in 15 minutes total. - Finish all 4 three-mark questions in 20 minutes total. - Tackle all 3 five-mark questions in 30 minutes (with brief working time). ✓ **HOTS Readiness:** - Review the case-study question above; create a similar scenario yourself (e.g., two forces on an inclined plane). - Practice integrating Vector Algebra with other topics (e.g., geometry proofs using vectors). Once all checkboxes are green, you're board-exam ready. Good luck!