Vector Algebra Class 12 Notes

• Points A, B, C are collinear if
• $\overrightarrow{AB}\times \overrightarrow{AC}=\vec{0}$
• (the displacement vectors are parallel).
• Vectors are coplanar if their scalar triple product is zero:
• $\vec{a}\cdot (\vec{b}\times \vec{c})=0$
• .
• Area of a parallelogram with sides
• $\vec{a},\vec{b}$
• is
• $|\vec{a}\times \vec{b}|$
• ; a triangle takes half of it.

Example: Find the angle between $\vec{a}=\hat{i}+2\hat{j}+2\hat{k}$ and $\vec{b}=3\hat{i}+0\hat{j}+4\hat{k}$.

Solution: $\vec{a}\cdot \vec{b}=3+0+8=11$, $|\vec{a}|=3$, $|\vec{b}|=5$, so $\cos \theta =11/15$ and $\theta ={\cos }^{-1}(11/15)$.

Example: Find the area of the triangle with vertices $(0,0,0),(1,2,3),(2,-1,1)$.

Solution: Take $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$ and $\vec{b}=2\hat{i}-\hat{j}+\hat{k}$. Then $\vec{a}\times \vec{b}=(5,5,-5)$ with length $5\sqrt{3}$, and the triangle area is half of that: $\frac{5\sqrt{3}}{2}$.

Exercises (NCERT Q1–Q54)

• 1) Represent graphically a displacement of 40 km, 30° west of north.
• 2) Classify the following measures as scalars and vectors: (i) 10 kg (ii) 2 metres north-west (iii) 40° (iv) 40 watt (v) 10⁻¹⁹ coulomb (vi) 20 m/s².
• 3) Classify the following as scalar and vector quantities: (i) time period (ii) distance (iii) force (iv) velocity (v) work done.
• 4) In the square figure (Fig 10.6) with centre O, the directed segments a, b, c and d are marked. Identify the vectors which are (i) coinitial, (ii) equal, (iii) collinear but not equal.
• 5) Answer the following as true or false: (i) a and −a are collinear. (ii) Two collinear vectors are always equal in magnitude. (iii) Two vectors having the same magnitude are collinear. (iv) Two collinear vectors having the same magnitude are equal.
• 6) Compute the magnitude of the following vectors: a = î + ĵ + k̂, b = 2î − 7ĵ − 3k̂ and c = (1/√3)î + (1/√3)ĵ − (1/√3)k̂.
• 7) Write two different vectors having the same magnitude.
• 8) Write two different vectors having the same direction.
• 9) Find the values of x and y so that the vectors 2î + 3ĵ and xî + yĵ are equal.
• 10) Find the scalar and vector components of the vector with initial point (2, 1) and terminal point (−5, 7).
• 11) Find the sum of the vectors a = î − 2ĵ + k̂, b = −2î + 4ĵ + 5k̂ and c = î − 6ĵ − 7k̂.
• 12) Find the unit vector in the direction of the vector a = î + ĵ + 2k̂.
• 13) Find the unit vector in the direction of the vector PQ, where P and Q are the points (1, 2, 3) and (4, 5, 6) respectively.
• 14) For given vectors a = 2î − ĵ + 2k̂ and b = −î + ĵ − k̂, find the unit vector in the direction of the vector a + b.
• 15) Find a vector in the direction of the vector 5î − ĵ + 2k̂ which has magnitude 8 units.
• 16) Show that the vectors 2î − 3ĵ + 4k̂ and −4î + 6ĵ − 8k̂ are collinear.
• 17) Find the direction cosines of the vector î + 2ĵ + 3k̂.
• 18) Find the direction cosines of the vector joining the points A (1, 2, −3) and B (−1, −2, 1), directed from A to B.
• 19) Show that the vector î + ĵ + k̂ is equally inclined to the axes OX, OY and OZ.
• 20) Find the position vector of a point R which divides the line joining the points P and Q, whose position vectors are î + 2ĵ − k̂ and −î + ĵ + k̂ respectively, in the ratio 2 : 1, (i) internally, (ii) externally.
• 21) Find the position vector of the mid point of the vector joining the points P (2, 3, 4) and Q (4, 1, −2).
• 22) Show that the points A, B and C with position vectors a = 3î − 4ĵ − 4k̂, b = 2î − ĵ + k̂ and c = î − 3ĵ − 5k̂ respectively form the vertices of a right angled triangle.
• 23) In triangle ABC, which of the following is not true? (A) AB + BC + CA = 0 (B) AB + BC − AC = 0 (C) AB + BC − CA = 0 (D) AB − CB + CA = 0.
• 24) If a and b are two collinear vectors, then which of the following are incorrect? (A) b = λa for some scalar λ (B) a = ± b (C) the respective components of a and b are proportional (D) both vectors a and b have the same direction, but different magnitudes.
• 25) Find the angle between two vectors a and b with magnitudes √3 and 2, respectively, having a·b = √6.
• 26) Find the angle between the vectors î − 2ĵ + 3k̂ and 3î − 2ĵ + k̂.
• 27) Find the projection of the vector î − ĵ on the vector î + ĵ.
• 28) Find the projection of the vector î + 3ĵ + 7k̂ on the vector 7î − ĵ + 8k̂.
• 29) Show that each of the given three vectors is a unit vector — (2î + 3ĵ + 6k̂)/7, (3î − 6ĵ + 2k̂)/7, (6î + 2ĵ − 3k̂)/7. Also show that they are mutually perpendicular to each other.
• 30) Find |a| and |b|, if (a + b)·(a − b) = 8 and |a| = 8|b|.
• 31) Evaluate the product (3a − 5b)·(2a + 7b).
• 32) Find the magnitude of two vectors a and b, having the same magnitude and such that the angle between them is 60° and their scalar product is 1/2.
• 33) Find |x|, if for a unit vector a, (x − a)·(x + a) = 12.
• 34) If a = 2î + 2ĵ + 3k̂, b = −î + 2ĵ + k̂ and c = 3î + ĵ are such that a + λb is perpendicular to c, then find the value of λ.
• 35) Show that |a| b + |b| a is perpendicular to |a| b − |b| a, for any two non-zero vectors a and b.
• 36) If a·a = 0 and a·b = 0, then what can be concluded about the vector b?
• 37) If a, b, c are unit vectors such that a + b + c = 0, find the value of a·b + b·c + c·a.
• 38) If either vector a = 0 or b = 0, then a·b = 0. But the converse need not be true. Justify your answer with an example.
• 39) If the vertices A, B, C of a triangle ABC are (1, 2, 3), (−1, 0, 0), (0, 1, 2), respectively, then find ∠ABC.
• 40) Show that the points A (1, 2, 7), B (2, 6, 3) and C (3, 10, −1) are collinear.
• 41) Show that the vectors 2î − ĵ + k̂, î − 3ĵ − 5k̂ and 3î − 4ĵ − 4k̂ form the vertices of a right angled triangle.
• 42) If a is a non-zero vector of magnitude 'a' and λ is a non-zero scalar, then λa is a unit vector if: (A) λ = 1 (B) λ = −1 (C) a = |λ| (D) a = 1/|λ|.
• 43) Find |a × b|, if a = î − 7ĵ + 7k̂ and b = 3î − 2ĵ + 2k̂.
• 44) Find a unit vector perpendicular to each of the vectors a + b and a − b, where a = 3î + 2ĵ + 2k̂ and b = î + 2ĵ − 2k̂.
• 45) If a unit vector a makes angles π/3 with î, π/4 with ĵ and an acute angle θ with k̂, then find θ and hence the components of a.
• 46) Show that (a − b) × (a + b) = 2(a × b).
• 47) Find λ and μ if (2î + 6ĵ + 27k̂) × (î + λĵ + μk̂) = 0.
• 48) Given that a·b = 0 and a × b = 0. What can you conclude about the vectors a and b?
• 49) Let the vectors a, b, c be given as a₁î + a₂ĵ + a₃k̂, b₁î + b₂ĵ + b₃k̂ and c₁î + c₂ĵ + c₃k̂ respectively. Showing the component-wise expansion, prove that a × (b + c) = a × b + a × c.
• 50) If either a = 0 or b = 0, then a × b = 0. Is the converse true? Justify your answer with an example.
• 51) Find the area of the triangle with vertices A (1, 1, 2), B (2, 3, 5) and C (1, 5, 5).
• 52) Find the area of the parallelogram whose adjacent sides are determined by the vectors a = î − ĵ + 3k̂ and b = 2î − 7ĵ + k̂.
• 53) Let the vectors a and b be such that |a| = 3 and |b| = √2/3. Then a × b is a unit vector if the angle between a and b is (A) π/6 (B) π/4 (C) π/3 (D) π/2.
• 54) Area of a rectangle having vertices A, B, C and D with position vectors −î + ½ĵ + 4k̂, î + ½ĵ + 4k̂, î − ½ĵ + 4k̂ and −î − ½ĵ + 4k̂ respectively is (A) ½ (B) 1 (C) 2 (D) 4.

Answers at a glance

1. 1
   Q1 → An arrow of 40 km magnitude pointing 30° west of north (drawn 4 cm long at scale 10 km = 1 cm).
2. 2
   Q2 → (i) scalar (ii) vector (iii) scalar (iv) scalar (v) scalar (vi) vector.
3. 3
   Q3 → (i) scalar (ii) scalar (iii) vector (iv) vector (v) scalar.
4. 4
   Q4 → (i) a and d; (ii) b and d; (iii) a and c.
5. 5
   Q5 → (i) True (ii) False (iii) False (iv) False.
6. 6
   Q6 → |a| = √3, |b| = √62 and |c| = 1.
7. 7
   Q7 → a = î + ĵ + k̂ and b = î + ĵ − k̂ (both of magnitude √3).
8. 8
   Q8 → a = î + 2ĵ + 3k̂ and b = 2î + 4ĵ + 6k̂ (b = 2a).
9. 9
   Q9 → x = 2, y = 3.
10. 10
    Q10 → Scalar components −7 and 6; vector components −7î and 6ĵ.
11. 11
    Q11 → a + b + c = −4ĵ − k̂.
12. 12
    Q12 → answer not in source — verify against the textbook
13. 13
    Q13 → answer not in source — verify against the textbook
14. 14
    Q14 → answer not in source — verify against the textbook
15. 15
    Q15 → answer not in source — verify against the textbook
16. 16
    Q16 → Collinear, since −4î + 6ĵ − 8k̂ = −2(2î − 3ĵ + 4k̂).
17. 17
    Q17 → l = 1/√14, m = 2/√14, n = 3/√14.
18. 18
    Q18 → Direction cosines are −1/3, −2/3, 2/3.
19. 19
    Q19 → All three direction cosines are 1/√3, so the vector is equally inclined to OX, OY, OZ.
20. 20
    Q20 → (i) (−î + 4ĵ + k̂)/3; (ii) −3î + 3k̂.
21. 21
    Q21 → 3î + 2ĵ + k̂ (point (3, 2, 1)).
22. 22
    Q22 → The triangle is right angled at A.
23. 23
    Q23 → Option (C).
24. 24
    Q24 → Incorrect: (B) and (D).
25. 25
    Q25 → answer not in source — verify against the textbook
26. 26
    Q26 → θ = cos⁻¹(5/7).
27. 27
    Q27 → 0.
28. 28
    Q28 → answer not in source — verify against the textbook
29. 29
    Q29 → Each is a unit vector and every pair has zero dot product — mutually perpendicular.
30. 30
    Q30 → answer not in source — verify against the textbook
31. 31
    Q31 → answer not in source — verify against the textbook
32. 32
    Q32 → |a| = |b| = 1.
33. 33
    Q33 → |x| = √13.
34. 34
    Q34 → λ = 8.
35. 35
    Q35 → The dot product is zero, so the vectors are perpendicular.
36. 36
    Q36 → a is the zero vector; b is arbitrary.
37. 37
    Q37 → answer not in source — verify against the textbook
38. 38
    Q38 → a = î and b = ĵ give a·b = 0 with a, b both non-zero — the converse fails.
39. 39
    Q39 → ∠ABC = cos⁻¹(10/√102).
40. 40
    Q40 → AB = BC, so A, B, C are collinear.
41. 41
    Q41 → The triangle is right angled at C.
42. 42
    Q42 → Option (D).
43. 43
    Q43 → 19√2.
44. 44
    Q44 → answer not in source — verify against the textbook
45. 45
    Q45 → answer not in source — verify against the textbook
46. 46
    Q46 → (a − b) × (a + b) = 0 + a×b + a×b − 0 = 2(a × b).
47. 47
    Q47 → answer not in source — verify against the textbook
48. 48
    Q48 → Either a = 0 or b = 0.
49. 49
    Q49 → a × (b + c) = a × b + a × c (right-distributive law verified).
50. 50
    Q50 → Converse is false: parallel non-zero vectors, e.g. î and 2î, have zero cross product.
51. 51
    Q51 → answer not in source — verify against the textbook
52. 52
    Q52 → Area = 15√2.
53. 53
    Q53 → Option (B).
54. 54
    Q54 → Option (C) — area 2.

MCQ Questions

1 Mark Questions

1. 1
   Write a unit vector in the direction of a = 3i + 4j. (Ans: (3i + 4j)/5)
2. 2
   When is the dot product of two non-zero vectors zero? (Ans: when they are perpendicular)
3. 3
   Give the magnitude of i × j. (Ans: 1)
4. 4
   State the condition for two vectors to be collinear. (Ans: one is a scalar multiple of the other)

2 Mark Questions

1. 1
   If a = 2i − j + k and b = i + 3j − 2k, find a·b. (Ans: 2 − 3 − 2 = −3)
2. 2
   Find the angle between a = i + j and b = i − j. (Ans: 90°)
3. 3
   Find a unit vector perpendicular to a = 2i + j and b = i + 2j. (Ans: ±k/√... (a × b)/|a × b| = ±k)

3 Mark Questions

1. 1
   Find the projection of a = 2i − j + k on b = i + 2j + 2k. (Ans: (a·b)/|b| = (2 − 2 + 2)/3 = 2/3)
2. 2
   Show that the vectors a = 3i − 2j + k, b = i − 3j + 5k, c = 2i + j − 4k form a right-angled triangle by computing a + b = ? Check endpoints and dot products. (Ans: a + b = 4i − 5j + 6k = c + ... identity verified)
3. 3
   If a, b are unit vectors inclined at 60°, find |a + b|. (Ans: √(1 + 1 + 2(1/2)) = √3)

4 Mark Questions

1. 1
   If a = 2i + 3j + k and b = 3i − j + 2k, find (a + b)·(a − b), |a + b|, |a − b| and the angle between a and b. Summarise which results use dot and which use algebra.
2. 2
   Prove the vector identity (a + b) × (a − b) = 2(b × a), then use it with a = 2i + j and b = i + 2j to find the area of the parallelogram they span. (Ans: area = |a × b| = 3 sq units)