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CAPF AC Vectors

Study Material — 1 PYQs (2019–2019) · Concept Notes · Shortcuts

CAPF AC Vectors is a frequently tested subtopic — 1 previous year questions from 2019–2019 papers are included below with concept notes, key rules and shortcut tricks.

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2019–2019
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Previous Year Questions

CAPF AC Vectors — Past Exam Questions

1 questions from actual CAPF AC papers · all shown free · click option to reveal solution

Exam Q 12019Previous Year Pattern

Let **a** = 2**i** + **j** + **k**, **b** = **i** − 2**j** + 3**k**, and **c** = 3**i** + **j** − 2**k** be three vectors. If **p** = **a** + λ**b** is perpendicular to **c**, then the value of λ is:

Concept Notes

Vectors— Rules & Concept

1.Core Concept

Read this first — the foundation of the topic

VECTORS — CORE CONCEPT A vector is a quantity that has both magnitude (size) and direction. For example, velocity is a vector — it tells you how fast AND which way. A scalar has only magnitude — like temperature or mass.

Vectors are written as bold letters (a) or with an arrow on top. The magnitude of vector a is written as |a|. ---

2.Key Rules

Core rules you must know cold
TYPES OF VECTORS

- Zero Vector: magnitude = 0, no fixed direction - Unit Vector: magnitude = 1. Written as â (a-hat) - Equal Vectors: same magnitude AND same direction - Negative Vector: same magnitude, opposite direction - Collinear Vectors: parallel to the same line - Position Vector: locates a point from the origin O 2

ADDITION OF VECTORS

- Triangle Law: Place vectors head to tail. The resultant goes from start to end. - Parallelogram Law: Two vectors from same point → diagonal = resultant - Commutative: a + b = b + a - Associative: (a + b) + c = a + (b + c) 3. DOT PRODUCT (Scalar Product): a · b = |a| |b| cos θ - Result is a SCALAR - If a · b = 0, vectors are perpendicular (θ = 90°) - a · a = |a|² 4. CROSS PRODUCT (Vector Product): |a × b| = |a| |b| sin θ - Result is a VECTOR - If a × b = 0, vectors are parallel (θ = 0° or 180°) - Direction is given by Right Hand Rule - a × b = −(b × a) — NOT commutative ---

3.Formula Block

Memorise — at least one formula appears in every paper
Unit vector â = a / |a|
If a = a1i + a2j + a3k, then |a| = sqrt(a1² + a2² + a3²)
Dot product: a · b = a1b1 + a2b2 + a3b3
Angle between vectors: cos θ = (a · b) / (|a| |b|)
Cross product magnitude: |a × b| = |a| |b| sin θ
Projection of a on b = (a · b) / |b|

---

4.Exam Patterns

What examiners ask — read before attempting PYQs
NDA regularly asks

- Finding angle between two vectors using dot product - Checking if vectors are perpendicular or parallel - Finding unit vectors - Magnitude of cross product (area of parallelogram) - Position vectors and section formula --- SHORTCUT / TRICK Trick 1 — Perpendicular Check: Just compute dot product. If a1b1 + a2b2 + a3b3 = 0, vectors are perpendicular. No need to find angle separately

Trick 2 — Area of Triangle

If two sides are given as vectors a and b, Area = (1/2)|a × b|

Remember

parallelogram area = |a × b|, triangle = half of that. ---

5.Worked Example

Solve this step-by-step before moving on

Question: Find the angle between vectors a = 2i + 2j − k and b = 6i − 3j + 2k. Step 1 — Find dot product: a · b = (2)(6) + (2)(−3) + (−1)(2) = 12 − 6 − 2 = 4 Step 2 — Find magnitudes: |a| = sqrt(4 + 4 + 1) = sqrt(9) = 3 |b| = sqrt(36 + 9 + 4) = sqrt(49) = 7 Step 3 — Apply formula: cos θ = 4 / (3 × 7) = 4/21 Step 4 — Answer: θ = cos⁻¹(4/21) ---

6.Exam Traps

Common mistakes students make — avoid these

Students confuse dot product and cross product results. Remember: Dot product gives a NUMBER (scalar). Cross product gives a VECTOR.

Also, many forget that a × b ≠ b × a — the sign flips.

Key Points to Remember

  • A vector has both magnitude and direction; a scalar has magnitude only.
  • Unit vector formula: â = a / |a|, and its magnitude is always 1.
  • Dot product formula: a · b = a1b1 + a2b2 + a3b3 (result is a scalar).
  • If dot product = 0, the two vectors are perpendicular to each other.
  • Cross product magnitude = |a||b|sin θ; if cross product = 0, vectors are parallel.
  • Area of parallelogram formed by vectors a and b = |a × b|.
  • Area of triangle formed by vectors a and b = (1/2)|a × b|.
  • Cross product is NOT commutative: a × b = −(b × a).

Exam-Specific Tips

  • The dot product of two perpendicular vectors is always zero (cos 90° = 0).
  • The cross product of two parallel vectors is always zero (sin 0° = 0).
  • If a = a1i + a2j + a3k, then |a| = sqrt(a1² + a2² + a3²).
  • Projection of vector a onto vector b = (a · b) / |b|.
  • The angle between vectors uses formula: cos θ = (a · b) / (|a| × |b|).
  • i × j = k, j × k = i, k × i = j (cyclic order for cross products of unit vectors).
  • i · i = j · j = k · k = 1, and i · j = j · k = k · i = 0.
Practice MCQs

Vectors — Practice Questions

38graded MCQs · easy to hard · full solution & trap analysis · showing 20 of 38

All MCQs →
Practice 1easy

The direction cosines of the vector **v** = 3**i** + 4**j** are:

Practice 2easy

Let **a** = 2**i** + 3**j** + **k** and **b** = **i** − 2**j** + 3**k**. The magnitude of the vector **a** × **b** is:

Practice 3easy

Two vectors **u** and **v** satisfy |**u**| = 5, |**v**| = 3, and **u** · **v** = 6. The angle θ between **u** and **v** is:

Practice 4easy

If **u** = 3**i** − **j** + 2**k** and **v** = **i** + 2**j** − **k**, then the dot product **u** · **v** equals:

Practice 5easy

Two vectors **p** and **q** are such that |**p**| = 4, |**q**| = 3, and **p** · **q** = 6. The angle θ between **p** and **q** is:

Practice 6easy

Let **a** = **i** + **j** + **k** and **b** = 2**i** − **j** + 3**k**. The cross product **a** × **b** is:

Practice 7easy

If **m** = 2**i** + **j** − **k** and **n** = **i** − 2**j** + 2**k**, then the magnitude of **m** × **n** is:

Practice 8easy

If **u** = 3**i** − **j** + 2**k** and **v** = **i** + 2**j** − **k**, then **u** · **v** equals:

Practice 9easy

The direction cosines of the vector **v** = 4**i** − 4**j** + 2**k** are:

Practice 10easy

Let **p** = 2**i** − **j** + **k** and **q** = **i** + 3**j** − 2**k**. The component of **p** along **q** is:

Practice 11easy

Let **a** = 2**i** + 3**j** + **k** and **b** = **i** − 2**j** + 3**k**. The magnitude of the vector **a** − **b** is:

Practice 12easy

If **a** = **i** + 2**j** + 3**k**, **b** = 2**i** − **j** + **k**, and **c** = **i** + **j** − **k**, then the scalar triple product **a** · (**b** × **c**) equals:

Practice 13easy

The angle θ between vectors **p** = **i** + **j** and **q** = **j** + **k** is given by cos θ =:

Practice 14easy

If **a** and **b** are two vectors such that |**a**| = 5, |**b**| = 3, and **a** · **b** = 12, then |**a** + **b**|² equals:

Practice 15medium

A line passes through the point P(1, 2, 3) and is parallel to the vector **d** = 2**i** + **j** − 2**k**. Find the distance from the point Q(2, 3, 4) to this line.

Practice 16medium

Let **a** = 2**i** + 3**j** + **k** and **b** = **i** − 2**j** + 3**k**. Find the magnitude of the vector **a** × **b**.

Practice 17medium

The vectors **a** = **i** + 2**j** + 3**k**, **b** = 2**i** + **j** − **k**, and **c** = 7**i** + 5**j** + **k** are coplanar if and only if their scalar triple product equals zero. Which of the following statements is correct?

Practice 18medium

Two lines in 3D space are given by: Line 1: **r** = (1, 2, 3) + t(2, 1, −1) Line 2: **r** = (4, 5, 2) + s(1, 2, 1) Find the angle θ between these two lines.

Practice 19medium

Let **u** = 3**i** − **j** + 2**k** and **v** = **i** + 2**j** − **k**. The vector **w** is perpendicular to both **u** and **v**. If **w** = **u** × **v**, then the unit vector in the direction of **w** is:

Practice 20medium

A plane passes through the point P(1, 2, 3) and has normal vector **n** = 2**i** + 3**j** − **k**. The equation of the plane is:

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60-Second Revision — Vectors

  • Remember: Dot product = scalar; Cross product = vector. Never mix these up.
  • Formula: cos θ = (a · b) / (|a||b|) — use this for angle between two vectors.
  • Trick: If a1b1 + a2b2 + a3b3 = 0, vectors are perpendicular — no extra steps needed.
  • Formula: Area of triangle = (1/2)|a × b|; Area of parallelogram = |a × b|.
  • Trap: Cross product is NOT commutative — a × b = −(b × a), sign changes.
  • Remember: Unit vector = original vector divided by its magnitude.
  • Trap: Parallel vectors have cross product = 0, NOT dot product = 0. Do not confuse the two conditions.
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