1. The cross product of two vectors gives a vector perpendicular to both vectors, with magnitude equal to the area of the parallelogram formed by the two vectors. 2. If two adjacent sides of a parallelogram are given by vectors a and b, the area of the parallelogram is |a x b|. 3. If the position vectors of three vertices of a triangle are given, the area of the triangle can be found as 1/2 times the magnitude of the cross product of any two sides of the triangle.

1. Vector Algebra
MATHEMATICS

2. Nishant Vora
B.Tech - IIT Patna
7+ years Teaching experience
Mentored 5 lac+ students
Teaching Excellence Award

3. Telegram Channel

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

9. Vector and
Scalar
Quantities

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

13. Laws of
Vector
Addition

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)

27. Magnitude of Vectors

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 ➝ ➝ ➝
➝ ➝ ➝ ➝

31. Collinear
Vectors

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

44. Properties
of Dot
Product

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
➝
➝ ➝ ➝ ➝ ➝
➝
➝
➝
➝
➝
➝
➝ ➝ ➝ ➝ ➝ ➝
➝
➝
➝
➝
➝
➝
➝ ➝ ➝ ➝ ➝ ➝
➝
➝
➝
➝
➝
➝

57. Questions on
Dot Product

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]

72. Components
of Vector

73. [JEE Main 2021]

75. [JEE Adv. 2015]

77. Linear
combination
of vectors

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

86. Properties of
Cross 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]

104. Geometrical
Interpretation of
Cross Product

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

110. A
D
C
B
Area of Quadrilateral

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]

114. Collinearity of
3 Points

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

121. Geometrical
Interpretation of
Box Product

122. Volume of Parallelepiped

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
➝ ➝ ➝
^ ^ ^ ^ ^ ^ ^

125. Properties of
Box Product

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]

138. Linearly
Dependent vs
Independent

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?
^ ^ ^ ^ ^ ^ ^ ^ ^

143. Vector Triple
Product (VTP)

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
➝ ➝ ➝ ➝ ➝ ➝ ➝
➝

161. Telegram Channel
tinyurl.com/specialclassNV10

162. Unacademy
Subscription
LIVE Polls & Leaderboard
LIVE Doubt Solving
LIVE Interaction
LIVE Class Environment Performance Analysis
Weekly Test Series
DPPs & Quizzes

163. India’s BEST Educators Unacademy Subscription
If you want to be the BEST
“Learn” from the BEST

164. Bratin Mondal
100 %ile
Amaiya singhal
100 %ile
Top
Results

165. NVLIVE
12th / Drop
11th / 9, 10
NVLIVE

167. Live Classes
Unlimited
Access
Structured
Courses
Weekly Tests
Personal Guidance
Get one on one guidance
from top exam experts
Test Analysis
Get one on one guidance
from top exam experts
Study Material
Specialised Notes & Practice
Sets
Study Planner
Customized study plan
with bi-weekly reviews
Experts' Guidelines
Study booster workshops by
exam experts
ICONIC PLUS

168. NVLIVE