![Properties of Dot Product
a . b =
5
➝ ➝
+
-
0
If θ ∈ [0, 𝝿/2)
If θ ∈ (𝝿/2, 𝝿]
If θ = 𝝿/2](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-49-320.jpg)
![Let a, b, c be three mutually perpendicular vectors of the same
magnitude and equally inclined at an angle θ, with the vector
a + b + c. Then 36 cos2 2θ is equal to _________.
[JEE Main 2021]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-58-320.jpg)
![Let a, b & c be three unit vectors such that |a - b|2 + |a - c|2 = 8. Then
|a + 2b|2 + |a + 2c|2 is equal to ________.
[JEE Main 2021]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-60-320.jpg)
![If a, b and c are unit vectors satisfying
|a - b|2 + |b - c|2 + |c - a|2 = 9, then |2a + 5b + 5c| is
➝➝ ➝
➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝
[JEE Adv. 2012]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-61-320.jpg)
![[JEE Main 2020]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-70-320.jpg)
![[JEE Main 2021]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-73-320.jpg)
![[JEE Adv. 2015]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-75-320.jpg)
![Let a = î + ĵ + k
̂ , b = î - ĵ + k
̂ and c = î - ĵ - k
̂ be there vectors. A vector
v in the plane of a and b, whose projection on c is 1/√3, is given by
➝ ➝ ➝
➝
A. î - 3ĵ + 3k
̂ B. -3î - 3ĵ - k
̂
C. 3î - ĵ + 3k
̂ D. î + ĵ - 3k
̂ [JEE 2011]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-81-320.jpg)
![[JEE Main 2021]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-82-320.jpg)
![[JEE Main 2021]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-98-320.jpg)
![[JEE Adv. 2011]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-99-320.jpg)
![[JEE Adv. 2020]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-100-320.jpg)
![Let △PQr be a triangle. Let a = QR, b = RP and c = PQ. if |a| = 12, |b|
= 4√3, b.c = 24, then which of the following is are) true?
A. B.
C. |a x b + c x a | = 48√3 D. a . b = -72
➝
➝
➝ __ __ __
➝ ➝ ➝
➝ ➝ ➝ ➝ ➝ ➝
[JEE Adv. 2015]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-102-320.jpg)
![[JEE Adv. 2020]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-112-320.jpg)
![[a b c] = -[a c b]
2
➝ ➝ ➝ ➝ ➝➝
Properties of Box Product (STP)](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-127-320.jpg)
![[a b b] = [a b a] = 0
4
➝➝➝ ➝➝➝
Properties of Box Product (STP)](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-129-320.jpg)
![The scalar triple product of three mutually perpendicular unit
vectors is ± 1. Thus [î ĵ k
̂ ] = 1, [î ĵ k
̂ ] = -1
5
Properties of Box Product (STP)](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-130-320.jpg)
![If two of the three vectors a, b, c are parallel then
[a b c] = 0
6 ➝ ➝ ➝
Properties of Box Product (STP)](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-131-320.jpg)
![Three non-zero non-collinear vectors are coplanar if
[a b c] = 0
7 ➝ ➝ ➝
Properties of Box Product (STP)](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-132-320.jpg)
![[a + b c d] = [a c d] + [b c d]
8
➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝
Properties of Box Product (STP)](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-133-320.jpg)
![[a + b b + c c + a] = 2[a b c]
9
➝
[a - b b - c c - a] = 0
10
[a ⨉ b b ⨉ c c ⨉ a] = [a b c]2
11
➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝
➝ ➝ ➝ ➝ ➝ ➝
➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝
➝
Properties of Box Product (STP)](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-134-320.jpg)
![For any three vectors a, b, c [a + b, b + c, c + a] equals
A. [a b c] B. 2 [a b c]
C. [a b c]2 D. 0
➝➝ ➝ ➝ ➝ ➝ ➝➝ ➝
➝➝➝
➝➝➝
➝➝➝](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-136-320.jpg)
![Let a, b and c be distinct positive numbers. If the vectors
then c is equal to
A. B.
C. D.
[JEE Main 2021]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-137-320.jpg)
![For 3 non zero Vectors
If [a b c] ≠ 0 ⇒ Linearly Independent
If [a b c] = 0 ⇒ Linearly Dependent
➝➝➝
➝ ➝➝
Linearly Dependent v/s Independent Vectors](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-140-320.jpg)
![[a ⨉ b b ⨉ c c ⨉ a] = [a b c]2
4
➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝
Vector Triple Product (VTP) - Properties](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-148-320.jpg)
![[JEE Main 2021]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-149-320.jpg)
![[JEE Main 2021]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-151-320.jpg)
![[JEE Adv. 2010]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-153-320.jpg)
![Vector Product of Four Vectors
V = (a ⨉ b) ⨉ (c ⨉ d) = [a b d]c - [a b c]d
➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝➝ ➝ ➝ ➝ ➝](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-156-320.jpg)
![If a, b, c and d are unit vectors such that (a x b) . (c x d) = 1 and
a . c = 1/ 2, then
➝ ➝ ➝ ➝ ➝ ➝ ➝
➝ ➝
A. a, b, c are non - coplanar
B. b, c, d are non - coplanar
C. b, d are non - parallel
D. a, d are parallel and b, c are parallel
➝ ➝ ➝
➝
➝
➝
➝ ➝
➝
➝
[JEE Adv. 2009]](https://image.slidesharecdn.com/vectoralgebraoneshotbounceback-240207124248-aa784c0d/85/Vector-Algebra-One-Shot-BounceBack-pdf-157-320.jpg)
Vector Algebra One Shot #BounceBack.pdf
- 2. Nishant Vora B.Tech - IIT Patna 7+ years Teaching experience Mentored 5 lac+ students Teaching Excellence Award
- 4. Unacademy Subscription LIVE Polls & Leaderboard LIVE Doubt Solving LIVE Interaction LIVE Class Environment Performance Analysis Weekly Test Series DPPs & Quizzes
- 5. India’s BEST Educators Unacademy Subscription If you want to be the BEST “Learn” from the BEST
- 6. Bratin Mondal 100 %ile Amaiya singhal 100 %ile Top Results
- 7. NVLIVE 12th / Drop 11th / 9, 10 NVLIVE
- 10. Physical Quantities Scalar Vector ➔ Only magnitude no direction ➔ Magnitude as well as direction and OBEYs vector law of algebra 1. Force 2. Velocity 3. Displacement 1. Mass 2. Volume 3. Density 4. Speed
- 11. Is Current a Vector Is is Is is current a vector Quantity? 3A 3A Current
- 12. Notation and Representation of Vectors Vectors are represented by a, b, c and their magnitude (modulus) are represented by a, b, c or |a|, |b|, |c|. ➝ ➝ ➝ ➝ ➝ ➝ Length of arrow ∝ magnitude
- 14. Addition of Vectors Triangle law of addition
- 15. Addition of Vectors Parallelogram law of addition
- 16. Addition of Vectors Loop law of addition
- 17. Addition of Vectors Loop law of addition x = a - b + c - d ➝ x = a + b + c + d x = a - b - c - d x = a - b + c + d ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ A. B. C. D. a b c d x
- 18. Addition of Vectors Loop law of addition x = a + b - c ➝ A. x = -a + b - c B. x = a - b - c C. x = -a - b + c D. ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ a b c x
- 19. Types of Vectors
- 20. Kinds of Vectors Zero or null vector 1 Magnitude = 0 Direction = Any arbitrary direction or No direction
- 21. Kinds of Vectors Magnitude = ➝ Direction = Unit vector in direction of a = Unit vector 2
- 22. Kinds of Vectors Unit vector 3 Unit vector in direction of x axis = Unit vector in direction of y axis = Unit vector in direction of z axis =
- 23. Kinds of Vectors Free vectors 4
- 24. Position Vectors Position vector of point P is __________ P ( 2, 1, 3)
- 25. Address of Point Cartesian System Vector System
- 26. Position Vectors P.V of point A = P.V of point B = AB = ➝ B (3, 2, 1) A ( 2, 1, 0)
- 28. Multiplication of a Vector by Scalar
- 29. Multiplication of a Vector by Scalar If a = xî + yĵ + zk ̂ Then ka = (kx)î + (ky)ĵ + (kz)k ̂ ➝ ➝
- 30. Properties of scalar multiplication k ( a + b ) = ka + kb 1 ( k + l ) a = k a + l a 2 ➝ ➝ ➝ ➝ ➝ ➝ ➝
- 32. Angle between two vectors Note: Tail to tail should be connected 0° ≤ 𝜃 ≤ 180°
- 33. Collinear Vectors or Parallel Vectors If angle between 2 vectors is either 0o or 180o 1
- 34. Collinear Vectors Collinear Vectors Like Vectors Unlike Vectors
- 35. Collinear Vectors or Parallel Vectors NOTE: If two vectors are parallel they are proportional NOTE: If two vectors are collinear then a = λ b
- 36. The value of λ when a = 2 - 3 + and b = 8î + λĵ + 4 are parallel is - A. 4 B. -6 C. -12 D. 1 ➝ ➝ ^ j ^ ^ i k
- 37. Coplanar Vectors Coplanar Vectors Vectors lying in same plane
- 38. Note: Two vectors are always coplanar 1 3 or more vectors may not be coplanar 2
- 39. Dot Product
- 40. Dot Product or Scalar Product Let a and b be two non-zero vectors and θ the angle between them then its scalar product is denoted as a . b and is defined as ➝ ➝ ➝ ➝ a . b = |a| |b| cos θ ➝ ➝ ➝ ➝
- 41. Dot Product or Scalar Product î · î = ĵ · ĵ = k ̂ . k ̂ = 1 1 î · ĵ = ĵ · k ̂ = k ̂ . î = 0 2
- 42. Dot Product or Scalar Product If a = a1î + a2ĵ + a3k ̂ and b = b1î + b2ĵ + b3k ̂ then a . b = a1b1 + a2b2 + a3b3 ➝ ➝ ➝ ➝
- 43. If a = 3i + 2j + k and b = i -2j + 5k then find a.b. ➝ ➝ ➝ ➝ A. 4 B. 5 D. -3 C. 3
- 45. Properties of Dot Product a . b = b . a 1 ➝ ➝ ➝ ➝
- 46. Properties of Dot Product a . a = |a|2 2 ➝ ➝ ➝
- 47. Properties of Dot Product a . (b + c) = a . b + a . c 3 ➝ ➝ ➝ ➝ ➝ ➝ ➝
- 48. Properties of Dot Product -|a| |b| ≤ a . b ≤ |a| |b| 4 ➝ ➝ ➝ ➝ ➝ ➝
- 49. Properties of Dot Product a . b = 5 ➝ ➝ + - 0 If θ ∈ [0, 𝝿/2) If θ ∈ (𝝿/2, 𝝿] If θ = 𝝿/2
- 50. Properties of Dot Product 0 . a = 0 6 ➝ ➝
- 51. Properties of Dot Product Angle between 2 vectors a and b cos θ = a.b/|a||b| 7 ➝ ➝ ➝ ➝
- 52. Properties of Dot Product If a ⟂ b then a . b = 0 but if a . b = 0 then either a = 0 or b = 0 or θ = 90° 8 ➝ ➝ ➝ ➝ ➝ ➝ ➝
- 53. Properties of Dot Product If b = c then a . b = a . c but if a . b = a . c then a = 0 or b = c or a ⟂ (b - c) 9 ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝
- 54. Properties of Dot Product Identities 10 a. (a + b)2 = |a|2 + 2 a . b + |b|2 b. (a - b)2 = |a|2 - 2 a . b + |b|2 c. (a + b) . (a - b) = |a|2 - |b|2 d. |a + b| = |a| + |b| ⇒ a || b e. |a + b|2 = |a|2 + |b|2 ⇒ a ⟂ b f. |a + b| = |a - b| ⇒ a ⟂ b ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝
- 58. Let a, b, c be three mutually perpendicular vectors of the same magnitude and equally inclined at an angle θ, with the vector a + b + c. Then 36 cos2 2θ is equal to _________. [JEE Main 2021]
- 60. Let a, b & c be three unit vectors such that |a - b|2 + |a - c|2 = 8. Then |a + 2b|2 + |a + 2c|2 is equal to ________. [JEE Main 2021]
- 61. If a, b and c are unit vectors satisfying |a - b|2 + |b - c|2 + |c - a|2 = 9, then |2a + 5b + 5c| is ➝➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ [JEE Adv. 2012]
- 63. Prove by vector method the following formula of trigonometry cos(α - β) = cosα cosβ + sinα sinβ #Power of Vectors
- 64. Projection of a on b
- 65. ➝ a ➝ b Projection of a on b
- 66. Projection of a on b
- 67. The projection of vector i + j + k on the vector i - j + k is- A. √3 B. 1/√3 C. 2/√3 D. 2√3 ^ ^ ^ ^ ^ ^
- 68. Angle between two vectors
- 69. Find the angle between the vectors 4î + ĵ + 3k ̂ and 2î - 2ĵ - k ̂ .
- 70. [JEE Main 2020]
- 73. [JEE Main 2021]
- 75. [JEE Adv. 2015]
- 78. A vector r is said to be a linear combination of the vectors a, b, c …. If ∃ scalars x, y, z, ….. Such that r = xa + yb + zc + …... ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ Linear combination of vectors
- 79. Theorem in Plane
- 80. Theorem in plane If three non-zero, non-collinear vectors are lying in same plane then any one vector can be represented as linear combination of other two
- 81. Let a = î + ĵ + k ̂ , b = î - ĵ + k ̂ and c = î - ĵ - k ̂ be there vectors. A vector v in the plane of a and b, whose projection on c is 1/√3, is given by ➝ ➝ ➝ ➝ A. î - 3ĵ + 3k ̂ B. -3î - 3ĵ - k ̂ C. 3î - ĵ + 3k ̂ D. î + ĵ - 3k ̂ [JEE 2011]
- 82. [JEE Main 2021]
- 84. Cross Product
- 85. The vector product or cross product of two vectors a and b is defined as a vector, written as a ⨉ b and is defined as ➝ ➝ ➝ ➝ a ⨉ b = |a| |b| sin θ n ^ ➝ ➝ ➝ ➝ θ n ^ ➝ a ⨉ b ➝ a b ➝ ➝ Cross product or Vector product
- 87. In general, a ⨉ b ≠ b ⨉ a. In fact a ⨉ b = -b ⨉ a 1 ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ Properties of Vector product
- 88. For scalar m, ma ⨉ b = m(a ⨉ b) = a ⨉ mb. 2 ➝ ➝ ➝ ➝ ➝ ➝ Properties of Vector product
- 89. a ⨉ (b ± c) = a ⨉ b ± a ⨉ c 3 ➝ ➝ ➝ ➝ ➝ ➝ ➝ Properties of Vector product
- 90. If a || b then θ = 0 or 𝝿 ⇒ a ⨉ b = 0 (but a ⨉ b = 0 ⇒ a = 0 or b = 0 or a || b). In particular a ⨉ a = 0. 4 ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ Properties of Vector product
- 91. If a ⟂ b then a ⨉ b = |a| |b|n (or |a ⨉ b| = |a| |b|) 5 ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ^ Properties of Vector product
- 92. î ⨉ î = ĵ ⨉ ĵ = k ̂ ⨉ k ̂ = 0 and î ⨉ ĵ = k ̂ , ĵ ⨉ k ̂ = î and k ̂ ⨉ î = ĵ (use cyclic system) 6 Properties of Vector product
- 93. Vector perpendicular to a and b is given by ± (a ⨉ b) 7 ➝ ➝ ➝ ➝ Properties of Vector product
- 94. If θ is angle between a and b then sin θ = |a ⨉ b|/|a||b| 8 ➝ ➝ ➝ ➝ ➝ ➝ Properties of Vector product
- 95. |a ⨉ b|2 + (a . b)2 = |a|2 |b|2 (Lagrange's identity) 9 ➝ ➝ ➝ ➝ ➝ ➝ Properties of Vector product
- 96. If a = a1î + a2ĵ + a3k ̂ and b = b1î + b2ĵ + b3k ̂ then 10 ➝ ➝ Properties of Vector product
- 97. If a = 2i + 2j - k and b = 6i - 3j + 2k then a x b equals A. 2i - 2j - k B. i - 10j - 18k C. i + j + k D. 6i - 3j + 2k ^ ➝ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ➝ ➝ ➝
- 98. [JEE Main 2021]
- 99. [JEE Adv. 2011]
- 100. [JEE Adv. 2020]
- 102. Let △PQr be a triangle. Let a = QR, b = RP and c = PQ. if |a| = 12, |b| = 4√3, b.c = 24, then which of the following is are) true? A. B. C. |a x b + c x a | = 48√3 D. a . b = -72 ➝ ➝ ➝ __ __ __ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ [JEE Adv. 2015]
- 105. If two adjacent sides are given CASE 1 Area of Parallelogram
- 106. If diagonal vectors are given CASE 2 Area of Parallelogram
- 107. n If position vectors of vertex are given CASE 3 Area of Parallelogram
- 108. If any two adjacent sides are given CASE 1 Area of Triangle
- 109. If position vectors of vertex are given CASE 2 Area of Triangle
- 111. Find the area of parallelogram whose two adjacent sides are represented by a = 3i + j + 2k and b = 2i - 2j + 4k. ^ ➝ ^ ^ ^ ^ ^ A. 8√3 B. 2√3 C. √3 D. 4√3
- 112. [JEE Adv. 2020]
- 115. Three points are collinear if ar(ABC)=0 or AB || BC || CA Collinearity of three points
- 116. If A ≡ (2i + 3j), B ≡ (pi + 9j) and C ≡ (i - j) are collinear, then the value of p is- ^ A. 1/2 B. 3/2 C. 7/2 D. 5/2 ^ ^ ^ ^ ^
- 117. Scalar Triple Product / Box Product
- 118. Scalar Triple Product or Box Product
- 119. If then Formula for Scalar Triple Product
- 123. A(a) C(c) B(b) 0 ➝ ➝ ➝ Volume of Tetrahedron
- 124. If a = 2i - 3j, b = i + j - k and c = 3i -k represent three coterminous edges of a parallelepiped then the volume of that parallelepiped is - A. 2 B. 4 C. 6 D. 10 ➝ ➝ ➝ ^ ^ ^ ^ ^ ^ ^
- 126. The position of (.) and (⨉) can be interchanged. i.e. a.(b ⨉ c) = (a ⨉ b).c 1 ➝ ➝ ➝ ➝ ➝ ➝ Properties of Box Product (STP)
- 127. [a b c] = -[a c b] 2 ➝ ➝ ➝ ➝ ➝➝ Properties of Box Product (STP)
- 128. You can rotate a, b, c in cyclic order 3 ➝➝➝ Properties of Box Product (STP)
- 129. [a b b] = [a b a] = 0 4 ➝➝➝ ➝➝➝ Properties of Box Product (STP)
- 130. The scalar triple product of three mutually perpendicular unit vectors is ± 1. Thus [î ĵ k ̂ ] = 1, [î ĵ k ̂ ] = -1 5 Properties of Box Product (STP)
- 131. If two of the three vectors a, b, c are parallel then [a b c] = 0 6 ➝ ➝ ➝ Properties of Box Product (STP)
- 132. Three non-zero non-collinear vectors are coplanar if [a b c] = 0 7 ➝ ➝ ➝ Properties of Box Product (STP)
- 133. [a + b c d] = [a c d] + [b c d] 8 ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ Properties of Box Product (STP)
- 134. [a + b b + c c + a] = 2[a b c] 9 ➝ [a - b b - c c - a] = 0 10 [a ⨉ b b ⨉ c c ⨉ a] = [a b c]2 11 ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ Properties of Box Product (STP)
- 135. ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ Properties of Box Product (STP)
- 136. For any three vectors a, b, c [a + b, b + c, c + a] equals A. [a b c] B. 2 [a b c] C. [a b c]2 D. 0 ➝➝ ➝ ➝ ➝ ➝ ➝➝ ➝ ➝➝➝ ➝➝➝ ➝➝➝
- 137. Let a, b and c be distinct positive numbers. If the vectors then c is equal to A. B. C. D. [JEE Main 2021]
- 139. For 2 non zero Vectors If a || b ⇒ Linearly Dependent If a ∦ b ⇒ Linearly Independent ➝ ➝ ➝ ➝ Linearly Dependent v/s Independent Vectors
- 140. For 3 non zero Vectors If [a b c] ≠ 0 ⇒ Linearly Independent If [a b c] = 0 ⇒ Linearly Dependent ➝➝➝ ➝ ➝➝ Linearly Dependent v/s Independent Vectors
- 141. 4 or more non zero Vectors Four or more vectors are always Linearly Dependent Linearly Dependent v/s Independent Vectors
- 142. Check whether i - 3j + 2k, 2i - 4j - k and 3i + 2j - k are linearly independent or dependent? ^ ^ ^ ^ ^ ^ ^ ^ ^
- 144. VTP - Definition
- 145. Vector triple product is a vector quantity 1 Vector Triple Product (VTP) - Properties
- 146. a ⨉ (b ⨉ c) ≠ (a ⨉ b) ⨉ c 2 ➝ ➝ ➝ ➝ ➝ ➝ Vector Triple Product (VTP) - Properties
- 147. (a ⨉ b) ⨉ c = 3 ➝ ➝ ➝ Vector Triple Product (VTP) - Properties
- 148. [a ⨉ b b ⨉ c c ⨉ a] = [a b c]2 4 ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ Vector Triple Product (VTP) - Properties
- 149. [JEE Main 2021]
- 151. [JEE Main 2021]
- 153. [JEE Adv. 2010]
- 155. Scalar Product of Four Vectors ➝ (a ⨉ b).(c ⨉ d) = ➝ ➝ ➝
- 156. Vector Product of Four Vectors V = (a ⨉ b) ⨉ (c ⨉ d) = [a b d]c - [a b c]d ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝➝ ➝ ➝ ➝ ➝
- 157. If a, b, c and d are unit vectors such that (a x b) . (c x d) = 1 and a . c = 1/ 2, then ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ A. a, b, c are non - coplanar B. b, c, d are non - coplanar C. b, d are non - parallel D. a, d are parallel and b, c are parallel ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝ [JEE Adv. 2009]
- 159. Isolating Unknown Vectors #Hathoda concept
- 160. Find vector r if r. a = m and r x b = c where a. b ≠ 0 ➝ ➝ ➝ ➝ ➝ ➝ ➝ ➝
- 162. Unacademy Subscription LIVE Polls & Leaderboard LIVE Doubt Solving LIVE Interaction LIVE Class Environment Performance Analysis Weekly Test Series DPPs & Quizzes
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