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Vectors formulas

Master Vectors through 36 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.

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Vectors, every formula

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Magnitude of a vector
$∣ a ∣ = x^{2} + y^{2} + z^{2}, a = x \hat{i} + y \hat{j} + z \hat{k}$
Unit vector
$\overset{a}{^} = \frac{a}{∣ a ∣}$
Q1MCQUnit vector
The unit vector along $\vec a=6\hat i+8\hat j$ is:
- A
  $\dfrac{3}{5}\hat i+\dfrac{4}{5}\hat j$
- B
  $6\hat i+8\hat j$
- C
  $\dfrac35\hat i+\dfrac35\hat j$
- D
  $\dfrac{1}{14}(6\hat i+8\hat j)$
Vector along AB
$A B = b - a$
position vectors a, b
Q1MCQVector AB
If $\vec a=(1,0,0)$ and $\vec b=(4,0,0)$, then $\vec{AB}$ is:
- A
  $(3,0,0)$
- B
  $(5,0,0)$
- C
  $(-3,0,0)$
- D
  $(1,0,0)$
Vector of magnitude m parallel to a
$\pm \frac{m a}{∣ a ∣}$
Q1MCQm-magnitude parallel
A vector of magnitude $10$ parallel to $\vec a=3\hat i+4\hat j$ is:
- A
  $6\hat i+8\hat j$
- B
  $3\hat i+4\hat j$
- C
  $30\hat i+40\hat j$
- D
  $\tfrac35\hat i+\tfrac45\hat j$
Section formula (internal)
$O C = \frac{m b + n a}{m + n}$
Q1MCQSection internal
The point dividing $\vec a,\vec b$ in ratio $1:1$ internally is:
- A
  $\dfrac{\vec a+\vec b}{2}$
- B
  $\vec a+\vec b$
- C
  $\dfrac{\vec b-\vec a}{2}$
- D
  $2(\vec a+\vec b)$
Section formula (external)
$O C = \frac{m b - n a}{m - n}$
Q1MCQSection external
The external-division position vector is:
- A
  $\dfrac{m\vec b-n\vec a}{m-n}$
- B
  $\dfrac{m\vec b+n\vec a}{m+n}$
- C
  $\dfrac{\vec a+\vec b}{2}$
- D
  $m\vec b-n\vec a$
Midpoint
$O C = \frac{a + b}{2}$
Q1MCQMidpoint
The midpoint of $\vec a$ and $\vec b$ is:
- A
  $\dfrac{\vec a+\vec b}{2}$
- B
  $\dfrac{\vec b-\vec a}{2}$
- C
  $\vec a+\vec b$
- D
  $\dfrac{\vec a+\vec b}{3}$
Centroid of a triangle
$G = \frac{a + b + c}{3}$
Q1MCQCentroid
The centroid of a triangle with position vectors $\vec a,\vec b,\vec c$ is:
- A
  $\dfrac{\vec a+\vec b+\vec c}{3}$
- B
  $\dfrac{\vec a+\vec b+\vec c}{2}$
- C
  $\vec a+\vec b+\vec c$
- D
  $\dfrac{\vec a+\vec b+\vec c}{4}$
Triangle inequality
$∣ a + b ∣ \leq ∣ a ∣ + ∣ b ∣$
Q1MCQTriangle inequality
For any two vectors, $|\vec a+\vec b|$ is:
- A
  $\le|\vec a|+|\vec b|$
- B
  $\ge|\vec a|+|\vec b|$
- C
  $=|\vec a|+|\vec b|$
- D
  $=|\vec a|-|\vec b|$
Collinear vectors
$a ∥ b \Leftrightarrow a = m b$
Q1MCQCollinear
$\vec a$ and $\vec b$ are collinear iff:
- A
  $\vec a=m\vec b$ for some scalar $m$
- B
  $\vec a\cdot\vec b=0$
- C
  $\vec a\times\vec b\neq0$
- D
  $|\vec a|=|\vec b|$
Collinearity of 3 points
$x a + y b + z c = 0, x + y + z = 0 (not all zero)$
Q1MCQCollinear points
Points with position vectors $\vec a,\vec b,\vec c$ are collinear iff there exist $x,y,z$ (not all zero) with $x\vec a+y\vec b+z\vec c=\vec 0$ and:
- A
  $x+y+z=0$
- B
  $xyz=0$
- C
  $x=y=z$
- D
  $x+y+z=1$
Coplanar vectors
$x a + y b + z c = 0 (not all zero)$
Q1MCQCoplanar
Three vectors are coplanar iff there exist scalars (not all zero) with:
- A
  $x\vec a+y\vec b+z\vec c=\vec 0$
- B
  $\vec a\cdot\vec b\cdot\vec c=0$
- C
  $x+y+z=1$
- D
  $\vec a\times\vec b\times\vec c=0$
Linear independence (3 vectors)
$Δ = a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3} \neq = 0$
Dot product (definition)
$a \cdot b = ∣ a ∣∣ b ∣ cos θ$
Dot product (components)
$a \cdot b = a_{1} b_{1} + a_{2} b_{2} + a_{3} b_{3}$
Angle via dot product
$cos θ = \frac{a \cdot b}{∣ a ∣∣ b ∣}$
Perpendicular vectors
$a \cdot b = 0 \Leftrightarrow a ⊥ b$
Q1MCQPerpendicular
$\vec a\cdot\vec b=0$ (both nonzero) means:
- A
  $\vec a\perp\vec b$
- B
  $\vec a\parallel\vec b$
- C
  $\vec a=\vec b$
- D
  $|\vec a|=|\vec b|$
Projection of b on a
$\frac{a \cdot b}{∣ a ∣}$
scalar projection
|a±b|²
$∣ a \pm b ∣^{2} = ∣ a ∣^{2} + ∣ b ∣^{2} \pm 2 a \cdot b$
Q1MCQ|a+b|²
$|\vec a+\vec b|^{2}$ equals:
- A
  $|\vec a|^{2}+|\vec b|^{2}+2\vec a\cdot\vec b$
- B
  $|\vec a|^{2}+|\vec b|^{2}$
- C
  $|\vec a|^{2}+|\vec b|^{2}-2\vec a\cdot\vec b$
- D
  $(|\vec a|+|\vec b|)^{2}$
Work done
$W = F \cdot S$
Cross product (definition)
$a \times b = ∣ a ∣∣ b ∣ sin θ \overset{n}{^}$
Q1MCQCross definition
$|\vec a\times\vec b|$ equals:
- A
  $|\vec a||\vec b|\sin\theta$
- B
  $|\vec a||\vec b|\cos\theta$
- C
  $|\vec a||\vec b|$
- D
  $|\vec a|+|\vec b|$
Cross product (determinant)
$a \times b = \hat{i} a_{1} b_{1} \hat{j} a_{2} b_{2} \hat{k} a_{3} b_{3}$
Q1MCQCross determinant
$\hat i\times\hat j$ equals:
- A
  $\hat k$
- B
  $-\hat k$
- C
  $\hat i$
- D
  $\vec 0$
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
  $\vec 0$
Parallel vectors (cross)
$a \times b = 0 \Leftrightarrow a ∥ b$
Q1MCQParallel (cross)
$\vec a\times\vec b=\vec 0$ (both nonzero) means:
- A
  $\vec a\parallel\vec b$
- B
  $\vec a\perp\vec b$
- C
  $\vec a=\vec b$
- D
  $\theta=90^{\circ}$
Unit normal to a and b
$\overset{n}{^} = \pm \frac{a \times b}{∣ a \times b ∣}$
Q1MCQUnit normal
A unit vector perpendicular to both $\vec a$ and $\vec b$ is:
- A
  $\pm\dfrac{\vec a\times\vec b}{|\vec a\times\vec b|}$
- B
  $\dfrac{\vec a\cdot\vec b}{|\vec a||\vec b|}$
- C
  $\vec a\times\vec b$
- D
  $\dfrac{\vec a+\vec b}{|\vec a+\vec b|}$
Area of a triangle
$Area = \frac{1}{2} ∣ a \times b ∣$
adjacent sides a, b
Area of a parallelogram
$Area = ∣ a \times b ∣$
Parallelogram via diagonals
$Area = \frac{1}{2} ∣ d_{1} \times d_{2} ∣$
Q1MCQParallelogram via diagonals
The area of a parallelogram in terms of diagonals $\vec d_1,\vec d_2$ is:
- A
  $\dfrac12|\vec d_1\times\vec d_2|$
- B
  $|\vec d_1\times\vec d_2|$
- C
  $\dfrac12\vec d_1\cdot\vec d_2$
- D
  $|\vec d_1||\vec d_2|$
Scalar triple product
$[a b c] = a \cdot (b \times c) = a_{1} b_{1} c_{1} a_{2} b_{2} c_{2} a_{3} b_{3} c_{3}$
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 (STP)
$[a b c] = 0 \Leftrightarrow coplanar$
Q1MCQSTP coplanarity
Three vectors are coplanar iff $[\vec a\ \vec b\ \vec c]$ equals:
- A
  $0$
- B
  $1$
- C
  $|\vec a||\vec b||\vec c|$
- D
  $\infty$
Volume of parallelepiped
$V = [a b c]$
Volume of tetrahedron
$V = \frac{1}{6} [a b c]$
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
  $(\vec a\times\vec b)\cdot\vec c$
- D
  $\vec a\cdot\vec b\cdot\vec c$
Lagrange's identity
$∣ a \times b ∣^{2} = ∣ a ∣^{2} ∣ b ∣^{2} - (a \cdot b)^{2}$
Q1MCQLagrange's identity
$|\vec a\times\vec b|^{2}$ equals:
- A
  $|\vec a|^{2}|\vec b|^{2}-(\vec a\cdot\vec b)^{2}$
- B
  $|\vec a|^{2}|\vec b|^{2}+(\vec a\cdot\vec b)^{2}$
- C
  $|\vec a|^{2}+|\vec b|^{2}$
- D
  $(\vec a\cdot\vec b)^{2}$
Vector eqn of a line
$r = a + t b$
through a, parallel to b
Q1MCQLine equation
The vector equation of a line through $\vec a$ parallel to $\vec b$ is:
- A
  $\vec r=\vec a+t\vec b$
- B
  $\vec r=\vec a\cdot\vec b$
- C
  $\vec r=t(\vec a+\vec b)$
- D
  $\vec r=\vec a\times\vec b$
i,j,k products
$\hat{i} \times \hat{j} = \hat{k}, \hat{j} \times \hat{k} = \hat{i}, \hat{k} \times \hat{i} = \hat{j}$
Q1MCQi,j,k product
$\hat j\times\hat k$ equals:
- A
  $\hat i$
- B
  $\hat k$
- C
  $-\hat i$
- D
  $\hat j$

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Vectors formulas

Vectors, every formula

Magnitude of a vector

Unit vector

Vector along AB

Vector of magnitude m parallel to a

Section formula (internal)

Section formula (external)

Midpoint

Centroid of a triangle

Triangle inequality

Collinear vectors

Collinearity of 3 points

Coplanar vectors

Linear independence (3 vectors)

Dot product (definition)

Dot product (components)

Angle via dot product

Perpendicular vectors

Projection of b on a

|a±b|²

Work done

Cross product (definition)

Cross product (determinant)

Anticommutativity

Parallel vectors (cross)

Unit normal to a and b

Area of a triangle

Area of a parallelogram

Parallelogram via diagonals

Scalar triple product

Coplanarity (STP)

Volume of parallelepiped

Volume of tetrahedron

Vector triple product

Lagrange's identity

Vector eqn of a line

i,j,k products

More JEE Advanced Mathematics formula sheets

Other ways to revise this chapter

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