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Vectors (Mathematics) formula sheet

Every formula in the Vectors (Mathematics) chapter for JEE Main, with one line on when it applies. Topic 1 is open to everyone; sign in free to read all 25 and download the PDF.

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How we count: 7× asked means 7 past JEE Main questions in the PrepWiser bank needed that formula, and "last" is the latest year one appeared. ★ marks formulas asked 8 or more times.

Vectors, position vectors and section formula

Magnitude and unit vector
4× asked, last 2024
a⃗=a1i^+a2j^+a3k^∣a⃗∣=a12+a22+a32,a^=a⃗∣a⃗∣\begin{aligned}&\vec a=a_1\hat i+a_2\hat j+a_3\hat k\\&|\vec a|=\sqrt{a_1^2+a_2^2+a_3^2},\qquad \hat a=\frac{\vec a}{|\vec a|}\end{aligned}
Collinear: b⃗=ka⃗\vec b=k\vec a. After squaring, keep only the roots that satisfy the original equation.
Section formula
5× asked, last 2026
AB→=b⃗−a⃗,p⃗=mb⃗+na⃗m+n  (AP:PB=m:n)\overrightarrow{AB}=\vec b-\vec a,\qquad \vec p=\frac{m\vec b+n\vec a}{m+n}\ \ (AP:PB=m:n)
Head minus tail. External: (mb⃗−na⃗)/(m−n)(m\vec b-n\vec a)/(m-n). Centroid (a⃗+b⃗+c⃗)/3(\vec a+\vec b+\vec c)/3.
Bisector meets the opposite side
3× asked, last 2024
BDDC=ABAC,d⃗=∣AC∣ b⃗+∣AB∣ c⃗∣AB∣+∣AC∣\frac{BD}{DC}=\frac{AB}{AC},\qquad \vec d=\frac{|AC|\,\vec b+|AB|\,\vec c}{|AB|+|AC|}
Each end is weighted by the other side. ∣OA∣=3|OA|=3, ∣OB∣=6|OB|=6: the bisector cuts ABAB at (2a⃗+b⃗)/3(2\vec a+\vec b)/3.
Triangle centres
3× asked, last 2025
g⃗=h⃗+2o⃗3;O at origin: h⃗=a⃗+b⃗+c⃗\vec g=\frac{\vec h+2\vec o}3;\qquad O\text{ at origin}:\ \vec h=\vec a+\vec b+\vec c
OO circumcentre, HH orthocentre, GG divides HOHO as 2:12:1. All coincide in an equilateral triangle.
Collinear points
4× asked, last 2024
A,B,C collinear  ⟺  AB→=λAC→  ⟺  AB→×AC→=0⃗\begin{aligned}&A,B,C\text{ collinear}\iff\overrightarrow{AB}=\lambda\overrightarrow{AC}\\&\iff\overrightarrow{AB}\times\overrightarrow{AC}=\vec0\end{aligned}
Equate components, starting with one that has no unknown.

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20 more formulas in 4 topics

Dot product, Cross product, Solving vector equations and Scalar triple product and coplanarity. Read the first part free. Sign in free with Google to read the rest and download the PDF.

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Dot product

  • Dot product and angle13×
  • Acute or obtuse4×
  • Projection and resolving23×
  • Squares of sums14×
  • Unit vectors adding to zero3×
  • Bounds3×
  • Angle bisector direction4×
  • Rotating a vector5×

Cross product

  • Cross product10×
  • Mixing a, b and a × b12×
  • Parallel vectors, common normal11×
  • Lagrange's identity13×
  • Areas19×
  • Cross product taken twice19×

Solving vector equations

  • Same vector crossed on both sides26×
  • Given a × x and a · x13×
  • In the plane of a and b11×
  • Coefficients of a combination8×

Scalar triple product and coplanarity

  • Scalar triple product31×
  • Tetrahedron

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