IIT JEE Test Series — Practice smarter, perform stronger.

Vectors formulas

Master Vectors through 30 JEE Advanced-level formulas, systematically structured with every variable spelled out. Revise concept-wise, identify the areas where you need improvement, and focus your preparation with greater precision.

Practise these formulas Know more about IIT JEE

Back to Formula Book

Vectors, every formula

30 formulas, typeset and free. Print it, or keep it open beside your practice.

Magnitude of a vector
$∣ A ∣ = A_{x}^{2} + A_{y}^{2} + A_{z}^{2}$
Unit vector
$\hat{A} = \frac{A}{∣ A ∣}$
Q1MCQUnit vector
The unit vector along $\vec A=6\hat i+8\hat j$ is:
- A
  $0.6\hat i+0.8\hat j$
- B
  $6\hat i+8\hat j$
- C
  $\hat i+\hat j$
- D
  $\dfrac{1}{14}(6\hat i+8\hat j)$
Components of a vector
$A_{x} = A cos θ, A_{y} = A sin θ$
Q1MCQComponents
The horizontal component of a vector $A$ at angle $\theta$ is:
- A
  $A\cos\theta$
- B
  $A\sin\theta$
- C
  $A\tan\theta$
- D
  $A$
Parallelogram law (resultant)
$R = A^{2} + B^{2} + 2 A B cos θ$
Direction of resultant
$tan α = \frac{B sin θ}{A + B cos θ}$
Q1MCQResultant direction
The direction of the resultant of $\vec A,\vec B$ (angle $\theta$) satisfies:
- A
  $\tan\alpha=\dfrac{B\sin\theta}{A+B\cos\theta}$
- B
  $\tan\alpha=\dfrac{B}{A}$
- C
  $\tan\alpha=\dfrac{A\sin\theta}{B}$
- D
  $\alpha=\theta$
Maximum resultant
$R_{m a x} = A + B (θ = 0)$
Minimum resultant
$R_{m i n} = ∣ A - B ∣ (θ = 18 0^{\circ})$
Perpendicular vectors' resultant
$R = A^{2} + B^{2} (θ = 9 0^{\circ})$
Q1MCQPerpendicular resultant
For two perpendicular vectors, the resultant magnitude is:
- A
  $\sqrt{A^{2}+B^{2}}$
- B
  $A+B$
- C
  $|A-B|$
- D
  $AB$
Position & displacement vector
$A B = b - a$
Q1MCQDisplacement vector
The vector from point $A$ to point $B$ is:
- A
  $\vec b-\vec a$
- B
  $\vec a-\vec b$
- C
  $\vec a+\vec b$
- D
  $\dfrac{\vec a+\vec b}{2}$
Distance between two points
$d = ∣ a - b ∣$
Section formula
$r = \frac{n a + m b}{m + n}$
Q1MCQSection formula
The point dividing $\vec a,\vec b$ internally in ratio $m:n$ is:
- A
  $\dfrac{n\vec a+m\vec b}{m+n}$
- B
  $\dfrac{m\vec a+n\vec b}{m+n}$
- C
  $\dfrac{\vec a+\vec b}{2}$
- D
  $m\vec b-n\vec a$
Midpoint
$r = \frac{a + b}{2}$
Q1MCQMidpoint
The midpoint of $\vec a$ and $\vec b$ is:
- A
  $\dfrac{\vec a+\vec b}{2}$
- B
  $\vec a+\vec b$
- C
  $\dfrac{\vec b-\vec a}{2}$
- D
  $\dfrac{\vec a+\vec b}{3}$
Dot product (definition)
$A \cdot B = A B cos θ$
Dot product (components)
$A \cdot B = A_{x} B_{x} + A_{y} B_{y} + A_{z} B_{z}$
Angle via dot product
$cos θ = \frac{A \cdot B}{A B}$
Q1MCQAngle
The angle between two vectors satisfies:
- A
  $\cos\theta=\dfrac{\vec A\cdot\vec B}{AB}$
- B
  $\sin\theta=\dfrac{\vec A\cdot\vec B}{AB}$
- C
  $\cos\theta=\vec A\cdot\vec B$
- D
  $\cos\theta=AB$
Perpendicular condition
$A \cdot B = 0 \Leftrightarrow A ⊥ B$
Q1MCQPerpendicular
Two nonzero vectors are perpendicular iff:
- A
  $\vec A\cdot\vec B=0$
- B
  $\vec A\times\vec B=0$
- C
  $\vec A=\vec B$
- D
  $A=B$
Projection
$proj = \frac{A \cdot B}{B}$
Unit-vector dot products
$\hat{i} \cdot \hat{i} = 1, \hat{i} \cdot \hat{j} = 0$
Cross product (definition)
$A \times B = A B sin θ \overset{n}{^}$
Q1MCQCross definition
$|\vec A\times\vec B|$ equals:
- A
  $AB\sin\theta$
- B
  $AB\cos\theta$
- C
  $AB$
- D
  $A+B$
Cross product (determinant)
$A \times B = \hat{i} A_{x} B_{x} \hat{j} A_{y} B_{y} \hat{k} A_{z} B_{z}$
Q1MCQCross determinant
$\hat i\times\hat j$ equals:
- A
  $\hat k$
- B
  $-\hat k$
- C
  $\hat i$
- D
  $0$
Magnitude of cross product
$∣ A \times B ∣ = area of parallelogram$
Q1MCQArea
$|\vec A\times\vec B|$ represents:
- A
  the area of the parallelogram of sides $\vec A,\vec B$
- B
  the dot product
- C
  the projection
- D
  the resultant
Parallel condition
$A \times B = 0 \Leftrightarrow A ∥ B$
Q1MCQParallel
Two nonzero vectors are parallel iff:
- A
  $\vec A\times\vec B=0$
- B
  $\vec A\cdot\vec B=0$
- C
  $A=B$
- D
  $\vec A=-\vec B$
Anticommutativity
$A \times B = - (B \times A)$
Q1MCQAnticommutativity
$\vec A\times\vec B$ equals:
- A
  $-(\vec B\times\vec A)$
- B
  $\vec B\times\vec A$
- C
  $\vec A\cdot\vec B$
- D
  $0$
Unit-vector cross products
$\hat{i} \times \hat{j} = \hat{k}, \hat{j} \times \hat{k} = \hat{i}, \hat{k} \times \hat{i} = \hat{j}$
Q1MCQCross j×k
$\hat j\times\hat k$ equals:
- A
  $\hat i$
- B
  $\hat k$
- C
  $-\hat i$
- D
  $\hat j$
Scalar triple product
$[A B C] = A \cdot (B \times C)$
Q1MCQScalar triple product
$[\vec A\ \vec B\ \vec C]$ equals:
- A
  $\vec A\cdot(\vec B\times\vec C)$
- B
  $\vec A\times(\vec B\cdot\vec C)$
- C
  $\vec A\cdot\vec B\cdot\vec C$
- D
  $(\vec A\times\vec B)\times\vec C$
Coplanarity
$[A B C] = 0$
Q1MCQCoplanarity
Three vectors are coplanar iff their scalar triple product is:
- A
  $0$
- B
  $1$
- C
  positive
- D
  $\infty$
Vector triple product
$A \times (B \times C) = (A \cdot C) B - (A \cdot B) C$
Q1MCQVector triple product
$\vec A\times(\vec B\times\vec C)$ equals:
- A
  $(\vec A\cdot\vec C)\vec B-(\vec A\cdot\vec B)\vec C$
- B
  $(\vec A\cdot\vec B)\vec C-(\vec A\cdot\vec C)\vec B$
- C
  $\vec A\cdot\vec B\cdot\vec C$
- D
  $0$
Lami's theorem
$\frac{A}{sin α} = \frac{B}{sin β} = \frac{C}{sin γ}$
three concurrent forces
Q1MCQLami's theorem
For three concurrent forces in equilibrium, Lami's theorem states:
- A
  $\dfrac{A}{\sin\alpha}=\dfrac{B}{\sin\beta}=\dfrac{C}{\sin\gamma}$
- B
  $A+B+C=0$
- C
  $A=B=C$
- D
  $\dfrac{A}{\cos\alpha}=\dfrac{B}{\cos\beta}$
|A+B|²
$∣ A + B ∣^{2} = A^{2} + B^{2} + 2 A B cos θ$
Q1MCQ|A+B|²
$|\vec A+\vec B|^{2}$ equals:
- A
  $A^{2}+B^{2}+2AB\cos\theta$
- B
  $A^{2}+B^{2}$
- C
  $A^{2}+B^{2}-2AB\cos\theta$
- D
  $(A+B)^{2}$
Work (dot product)
$W = F \cdot d$

Other ways to revise this chapter

Master this chapter with similar other learning materials.

Preparing students for India’s top institutes

Our students are currently into top technological and medical institutes of India.

IIT Bombay
IIT Delhi
IIT Madras
IIT Kanpur
IIT Kharagpur
IIT Roorkee
IIT Guwahati
IIT BHU Varanasi
AIIMS Delhi
NIT Tiruchirappalli
NIT Rourkela
IIT Bombay
IIT Delhi
IIT Madras
IIT Kanpur
IIT Kharagpur
IIT Roorkee
IIT Guwahati
IIT BHU Varanasi
AIIMS Delhi
NIT Tiruchirappalli
NIT Rourkela

Join QuestPix, Today!

Get notified first, with exam & curriculum updates, course & test series launch offers, motivation & success stories and free learning resources recommended by toppers.

Vectors formulas

Vectors, every formula

Magnitude of a vector

Unit vector

Components of a vector

Parallelogram law (resultant)

Direction of resultant

Maximum resultant

Minimum resultant

Perpendicular vectors' resultant

Position & displacement vector

Distance between two points

Section formula

Midpoint

Dot product (definition)

Dot product (components)

Angle via dot product

Perpendicular condition

Projection

Unit-vector dot products

Cross product (definition)

Cross product (determinant)

Magnitude of cross product

Parallel condition

Anticommutativity

Unit-vector cross products

Scalar triple product

Coplanarity

Vector triple product

Lami's theorem

|A+B|²

Work (dot product)

More JEE Advanced Physics formula sheets

Other ways to revise this chapter

Preparing students for India’s top institutes

Join QuestPix, Today!